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<div id="doc" class="markdown-body container-fluid"><h1 id="stats-2-notes"><a class="anchor hidden-xs" href="#stats-2-notes" title="stats-2-notes"><span class="octicon octicon-link"></span></a>Stats 2 notes</h1><h2 id="discrete-random-variables"><a class="anchor hidden-xs" href="#discrete-random-variables" title="discrete-random-variables"><span class="octicon octicon-link"></span></a>Discrete random variables</h2><p><strong>Discrete random variable</strong> A random variable for which a list of all possible values could be made</p><p><strong>Probability distribution</strong> A list or table showing the probability of each value occurring</p><p>The sum of the probabilities in the probability distribution equals <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-116-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1613" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1614"><span class="mn" id="MathJax-Span-1615" style="font-family: STIXGeneral;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-116">1</script></span></p><p><strong>Probability function</strong> A function which provides <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-117-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1616" style="width: 4.511em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.691em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.64em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1617"><span class="mi" id="MathJax-Span-1618" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-1619" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1620" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1621" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1622" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1623" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-117">P(X = x)</script></span> for all <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-118-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1624" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1625"><span class="mi" id="MathJax-Span-1626" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-118">x</script></span></p><p><strong>Cumulative distribution function</strong> A function which provides <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-119-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&#x2264;</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1627" style="width: 4.511em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.691em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.64em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1628"><span class="mi" id="MathJax-Span-1629" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-1630" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1631" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1632" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mi" id="MathJax-Span-1633" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1634" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-119">P(X \leq x)</script></span> for all <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-120-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1635" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1636"><span class="mi" id="MathJax-Span-1637" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-120">x</script></span></p><p>Mean <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-231-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2211;</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3225" style="width: 9.941em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.148em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1008.15em 2.974em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3226"><span class="mi" id="MathJax-Span-3227" style="font-family: STIXGeneral; font-style: italic;">μ</span><span class="mo" id="MathJax-Span-3228" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-3229" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-3230" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-3231"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3232" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.72em;"><span class="mn" id="MathJax-Span-3233" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3234" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3235" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-3236" style="font-family: STIXGeneral; vertical-align: 0.003em; padding-left: 0.31em;">∑</span><span class="msubsup" id="MathJax-Span-3237" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3238" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-3239" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-3240"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3241" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-3242" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.372em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-231">\mu = E(X^2) = \sum x_i p_i</script></span> (The expectation of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-232-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3243" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3244"><span class="mi" id="MathJax-Span-3245" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-232">X</script></span>)</p><p>Variance <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-122-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&#x03C3;</mi><mn>2</mn></msup><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mo>&#x2211;</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><msub><mi>p</mi><mi>i</mi></msub><mo>&#x2212;</mo><mo>&#x2211;</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1641" style="width: 19.828em; display: inline-block;"><span style="display: inline-block; position: relative; width: 16.242em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1016.24em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1642"><span class="msubsup" id="MathJax-Span-1643"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1644" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-1645" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1646" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1647" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1648" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1649"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1650" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.72em;"><span class="mn" id="MathJax-Span-1651" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1652" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1653" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1654" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1655" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1656" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-1657"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-1658" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.361em;"><span class="mn" id="MathJax-Span-1659" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1660" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-1661" style="font-family: STIXGeneral; vertical-align: 0.003em; padding-left: 0.31em;">∑</span><span class="msubsup" id="MathJax-Span-1662" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1663" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.515em;"><span class="mn" id="MathJax-Span-1664" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -3.686em; left: 0.464em;"><span class="mi" id="MathJax-Span-1665" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-1666"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1667" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1668" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1669" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mo" id="MathJax-Span-1670" style="font-family: STIXGeneral; vertical-align: 0.003em; padding-left: 0.259em;">∑</span><span class="msubsup" id="MathJax-Span-1671" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1672" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-1673" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-1674"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1675" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1676" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>−</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><msub><mi>p</mi><mi>i</mi></msub><mo>−</mo><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-122">\sigma ^2 = E(X^2) - E(X)^2 = \sum x_i ^2 p_i - \sum x_i p_i</script></span></p><h3 id="expectation-of-a-function-of-a-random-variable"><a class="anchor hidden-xs" href="#expectation-of-a-function-of-a-random-variable" title="expectation-of-a-function-of-a-random-variable"><span class="octicon octicon-link"></span></a>Expectation of a function of a random variable</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-123-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2211;</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mi>p</mi><mi>i</mi></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1677" style="width: 10.453em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.558em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.847em 1008.56em 3.589em -999.997em); top: -2.969em; left: 0em;"><span class="mrow" id="MathJax-Span-1678"><span class="mi" id="MathJax-Span-1679" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1680" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1681" style="font-family: STIXGeneral; font-style: italic;">g</span><span class="mo" id="MathJax-Span-1682" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1683" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1684" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1685" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1686" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-1687" style="font-family: STIXSizeOneSym; vertical-align: -0.51em; padding-left: 0.31em;">∑</span><span class="mi" id="MathJax-Span-1688" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">g</span><span class="mo" id="MathJax-Span-1689" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1690"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1691" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-1692" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1693" style="font-family: STIXGeneral;">)</span><span class="msubsup" id="MathJax-Span-1694"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1695" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1696" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.974em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.622em; border-left: 0px solid; width: 0px; height: 1.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo>∑</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mi>p</mi><mi>i</mi></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-123"> E(g(X)) = \sum g(x_i)p_i </script></span></p><h3 id="mean-and-variance-of-functions-of-a-random-variable"><a class="anchor hidden-xs" href="#mean-and-variance-of-functions-of-a-random-variable" title="mean-and-variance-of-functions-of-a-random-variable"><span class="octicon octicon-link"></span></a>Mean and variance of functions of a random variable</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-124-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>E</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em" /><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mspace width="1em" /><mi>E</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1697" style="width: 32.226em; display: inline-block;"><span style="display: inline-block; position: relative; width: 26.386em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1026.34em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1698"><span class="mi" id="MathJax-Span-1699" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1700" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1701" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1702" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1703" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1704" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1705" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">a</span><span class="mi" id="MathJax-Span-1706" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1707" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1708" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1709" style="font-family: STIXGeneral;">)</span><span class="mspace" id="MathJax-Span-1710" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-1711" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1712" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1713" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1714" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-1715" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mo" id="MathJax-Span-1716" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1717" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1718" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1719" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1720" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1721" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1722" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-1723" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mspace" id="MathJax-Span-1724" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-1725" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1726" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1727" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1728" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1729" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-1730" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mo" id="MathJax-Span-1731" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1732" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1733" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">a</span><span class="mi" id="MathJax-Span-1734" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1735" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1736" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1737" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1738" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-1739" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>E</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mspace width="1em"></mspace><mi>E</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-124"> E(aX) = aE(x) \quad E(X+b) = E(X) + b \quad E(aX + b) = aE(X) + b</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-125-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em" /><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="1em" /><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1740" style="width: 36.376em; display: inline-block;"><span style="display: inline-block; position: relative; width: 29.818em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1029.77em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1741"><span class="mi" id="MathJax-Span-1742" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-1743" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1744" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1745" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1746" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1747" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1748" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1749" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-1750" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1751" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mn" id="MathJax-Span-1752" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-1753" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-1754" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1755" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1756" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1757" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1758" style="font-family: STIXGeneral;">)</span><span class="mspace" id="MathJax-Span-1759" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-1760" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; 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padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1769" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-1770" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1771" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1772" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1773" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1774" style="font-family: STIXGeneral;">)</span><span class="mspace" id="MathJax-Span-1775" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; 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font-style: italic; padding-left: 0.259em;">b</span><span class="mo" id="MathJax-Span-1784" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1785" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-1786" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1787" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mn" id="MathJax-Span-1788" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-1789" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-1790" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1791" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1792" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1793" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1794" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.441em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-125">Var(aX) = a^2Var(x) \quad Var(X+b) = Var(X) \quad Var(aX+b) = a^2Var(X)</script></span></p><h2 id="the-poisson-distribution"><a class="anchor hidden-xs" href="#the-poisson-distribution" title="the-poisson-distribution"><span class="octicon octicon-link"></span></a>The poisson distribution</h2><h3 id="conditions"><a class="anchor hidden-xs" href="#conditions" title="conditions"><span class="octicon octicon-link"></span></a>Conditions</h3><ul>
<li>The time or position of each event is independent of previous events</li>
<li>The probability of each event occurring in a given time interval or space if fixed</li>
<li>Two events cannot occur at exactly the same time or in the same position</li>
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display: inline-block;"><span style="display: inline-block; position: relative; width: 14.808em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.205em 1014.81em 5.587em -999.997em); top: -4.147em; left: 0em;"><span class="mrow" id="MathJax-Span-1796"><span class="mi" id="MathJax-Span-1797" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-1798" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1799" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1800" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1801" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1802" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1803" style="font-family: STIXGeneral; 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position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="texatom" id="MathJax-Span-1828"><span class="mrow" id="MathJax-Span-1829"><span class="mi" id="MathJax-Span-1830" style="font-family: STIXGeneral;">ℕ</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.72em;"><span class="mn" id="MathJax-Span-1831" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1005.02em 4.152em -999.997em); top: -3.071em; left: 0em;"><span class="mtd" id="MathJax-Span-1835"><span class="mrow" id="MathJax-Span-1836"><span class="mspace" id="MathJax-Span-1837" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-1838" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-1839" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-1840" style="font-family: STIXGeneral; font-style: italic;">h</span><span class="mi" id="MathJax-Span-1841" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mi" id="MathJax-Span-1842" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-1843" style="font-family: STIXGeneral; font-style: italic;">w</span><span class="mi" id="MathJax-Span-1844" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-1845" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-1846" style="font-family: STIXGeneral; font-style: italic;">e</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1847"></span></span></span><span style="display: inline-block; width: 0px; height: 4.152em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.622em; border-left: 0px solid; width: 0px; height: 3.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>λ</mi></mrow></msup><mfrac><msup><mi>λ</mi><mi>x</mi></msup><mrow><mi>x</mi><mo>!</mo></mrow></mfrac></mtd><mtd><mspace width="1em"></mspace><mi>x</mi><mo>∈</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow><mn>0</mn></msup></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mspace width="1em"></mspace><mi>o</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>w</mi><mi>i</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-126">P(X=x) = \left\{
\begin{array}{ll}
e^{-\lambda} \frac{\lambda^x}{x!} & \quad x \in \mathbb{N}^0 \\
0 & \quad otherwise
\end{array}
\right.</script></span></p><h3 id="the-recurrence-formula"><a class="anchor hidden-xs" href="#the-recurrence-formula" title="the-recurrence-formula"><span class="octicon octicon-link"></span></a>The recurrence formula</h3><p>In order to calculate the succession of values of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-127-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1848" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1849"><span class="mi" id="MathJax-Span-1850" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-127">x</script></span>: <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-128-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1851" style="width: 11.632em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.531em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1009.48em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1852"><span class="mi" id="MathJax-Span-1853" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-1854" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1855" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1856" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-1857" style="font-family: STIXGeneral; padding-left: 0.31em;">1</span><span class="mo" id="MathJax-Span-1858" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1859" style="font-family: STIXGeneral;">,</span><span class="mi" id="MathJax-Span-1860" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">P</span><span class="mo" id="MathJax-Span-1861" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1862" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1863" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-1864" style="font-family: STIXGeneral; padding-left: 0.31em;">2</span><span class="mo" id="MathJax-Span-1865" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1866" style="font-family: STIXGeneral;">,</span><span class="mo" id="MathJax-Span-1867" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-1868" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-1869" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-128">P(X=1), P(X=2), ...</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-129-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>&#x03BB;</mi><mi>x</mi></mfrac><mo>&#x00D7;</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1870" style="width: 15.013em; display: inline-block;"><span style="display: inline-block; position: relative; width: 12.298em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1012.25em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1871"><span class="mi" id="MathJax-Span-1872" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-1873" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1874" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1875" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-1876" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1877" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-1878" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1879" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1880" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1881" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1882" style="font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1883" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.57em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.566em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1884" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mi" id="MathJax-Span-1885" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">P</span><span class="mo" id="MathJax-Span-1886" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1887" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1888" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-1889" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.693em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1890" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1891"><span class="mrow" id="MathJax-Span-1892"><span class="mi" id="MathJax-Span-1893" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1894" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1895" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1896" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>λ</mi><mi>x</mi></mfrac><mo>×</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-129">P(X= x_n) = \frac{\lambda}{x} \times P(X = x_{n-1})</script></span></p><h3 id="sum-of-independent-random-variables"><a class="anchor hidden-xs" href="#sum-of-independent-random-variables" title="sum-of-independent-random-variables"><span class="octicon octicon-link"></span></a>Sum of independent random variables</h3><p>Two or more independent Poisson distributions can be combined as follows</p><p>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-237-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub><mo>&#x223C;</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>&#x03BB;</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>&#x223C;</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>&#x03BB;</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>X</mi><mi>n</mi></msub><mo>&#x223C;</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>&#x03BB;</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3388" style="width: 19.726em; display: inline-block;"><span style="display: inline-block; position: relative; width: 16.14em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1016.09em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3389"><span class="msubsup" id="MathJax-Span-3390"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3391" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-3392" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3393" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-3394" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">P</span><span class="mo" id="MathJax-Span-3395" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-3396"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3397" style="font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-3398" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3399" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3400" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-3401" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3402" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-3403" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3404" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-3405" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">P</span><span class="mo" id="MathJax-Span-3406" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-3407"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3408" style="font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-3409" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3410" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3411" style="font-family: STIXGeneral;">,</span><span class="mo" id="MathJax-Span-3412" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-3413" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-3414" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-3415" style="font-family: STIXGeneral; padding-left: 0.207em;">,</span><span class="msubsup" id="MathJax-Span-3416" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3417" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-3418" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3419" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-3420" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">P</span><span class="mo" id="MathJax-Span-3421" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-3422"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3423" style="font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-3424" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3425" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>X</mi><mi>n</mi></msub><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-237">X_1 \sim P(\lambda _1), X_2 \sim P(\lambda _2), ..., X_n \sim P(\lambda _n)</script></span> then <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-238-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>X</mi><mi>k</mi></msub><mo>&#x223C;</mo><mi>P</mi><mo stretchy="false">(</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>&#x03BB;</mi><mi>k</mi></msub><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3426" style="width: 9.07em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.431em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1007.38em 3.998em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3427"><span class="munderover" id="MathJax-Span-3428"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.18em 4.613em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-3429" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -2.866em; left: 0.054em;"><span class="texatom" id="MathJax-Span-3430"><span class="mrow" id="MathJax-Span-3431"><span class="mi" id="MathJax-Span-3432" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-3433" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-3434" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.36em 4.152em -999.997em); top: -5.172em; left: 0.464em;"><span class="mi" id="MathJax-Span-3435" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-3436" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3437" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-3438" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3439" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-3440" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">P</span><span class="mo" id="MathJax-Span-3441" style="font-family: STIXGeneral;">(</span><span class="munderover" id="MathJax-Span-3442"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.18em 4.613em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-3443" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -2.866em; left: 0.054em;"><span class="texatom" id="MathJax-Span-3444"><span class="mrow" id="MathJax-Span-3445"><span class="mi" id="MathJax-Span-3446" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-3447" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-3448" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.36em 4.152em -999.997em); top: -5.172em; left: 0.464em;"><span class="mi" id="MathJax-Span-3449" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-3450" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3451" style="font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-3452" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3453" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.622em; border-left: 0px solid; width: 0px; height: 3.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>X</mi><mi>k</mi></msub><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>λ</mi><mi>k</mi></msub><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-238">\sum _{k=1} ^n X_k \sim P(\sum _{k=1} ^n \lambda _k )</script></span></p><p>This also shows that if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-247-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>&#x223C;</mo><mi>P</mi><mo stretchy="false">(</mo><mi>&#x03BB;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3526" style="width: 4.511em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.691em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.64em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3527"><span class="mi" id="MathJax-Span-3528" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-3529" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-3530" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">P</span><span class="mo" id="MathJax-Span-3531" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3532" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="mo" id="MathJax-Span-3533" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-247">X \sim P(\lambda)</script></span> then</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-248-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>X</mi><mo>&#x223C;</mo><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mi>&#x03BB;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3534" style="width: 5.791em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.716em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1004.66em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3535"><span class="mi" id="MathJax-Span-3536" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-3537" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-3538" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-3539" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">P</span><span class="mo" id="MathJax-Span-3540" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3541" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-3542" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="mo" id="MathJax-Span-3543" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>X</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mi>λ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-248">nX \sim P(n \lambda)</script></span></p><p>Example</p><blockquote>
<p>At a checkpoint an average of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-249-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3544" style="width: 1.829em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.501em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1001.45em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3545"><span class="mn" id="MathJax-Span-3546" style="font-family: STIXGeneral;">300</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn></math></span></span><script type="math/tex" id="MathJax-Element-249">300</script></span> cars pass per hour and the mean time between lorries is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-250-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3547" style="width: 0.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1000.47em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3548"><span class="mn" id="MathJax-Span-3549" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-250">5</script></span> minutes.<br>
Find the probability that exactly <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-251-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3550" style="width: 0.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1000.47em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3551"><span class="mn" id="MathJax-Span-3552" style="font-family: STIXGeneral;">6</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-251">6</script></span> vehicles pass the checkpoint in a <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-252-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3553" style="width: 0.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1000.42em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3554"><span class="mn" id="MathJax-Span-3555" style="font-family: STIXGeneral;">1</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.917em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-252">1</script></span> minute period.<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-253-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3556" style="width: 1.829em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.501em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1001.45em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3557"><span class="mn" id="MathJax-Span-3558" style="font-family: STIXGeneral;">300</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn></math></span></span><script type="math/tex" id="MathJax-Element-253">300</script></span> cars per hour <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-254-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">&#x2192;</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3559" style="width: 1.173em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.923em 1000.89em 2.625em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3560"><span class="mo" id="MathJax-Span-3561" style="font-family: STIXGeneral;">→</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: 0.003em; border-left: 0px solid; width: 0px; height: 0.574em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">→</mo></math></span></span><script type="math/tex" id="MathJax-Element-254">\to</script></span> <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-255-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3562" style="width: 0.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1000.47em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3563"><span class="mn" id="MathJax-Span-3564" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-255">5</script></span> cars per minute<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-256-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3565" style="width: 0.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1000.42em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3566"><span class="mn" id="MathJax-Span-3567" style="font-family: STIXGeneral;">1</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.917em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-256">1</script></span> lorry per <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-257-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3568" style="width: 0.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1000.47em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3569"><span class="mn" id="MathJax-Span-3570" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-257">5</script></span> minutes <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-258-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">&#x2192;</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3571" style="width: 1.173em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.923em 1000.89em 2.625em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3572"><span class="mo" id="MathJax-Span-3573" style="font-family: STIXGeneral;">→</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: 0.003em; border-left: 0px solid; width: 0px; height: 0.574em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">→</mo></math></span></span><script type="math/tex" id="MathJax-Element-258">\to</script></span> <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-259-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.2</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3574" style="width: 1.548em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.267em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1001.22em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3575"><span class="mn" id="MathJax-Span-3576" style="font-family: STIXGeneral;">0.2</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.2</mn></math></span></span><script type="math/tex" id="MathJax-Element-259">0.2</script></span> lorries per minute</p>
<p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-260-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5.2</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3577" style="width: 1.548em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.267em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1001.22em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3578"><span class="mn" id="MathJax-Span-3579" style="font-family: STIXGeneral;">5.2</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5.2</mn></math></span></span><script type="math/tex" id="MathJax-Element-260">5.2</script></span> vehicles per minute<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-261-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>6</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>5.2</mn></mrow></msup><mo>&#x00D7;</mo><mfrac><msup><mn>5.2</mn><mn>6</mn></msup><mrow><mn>6</mn><mo>!</mo></mrow></mfrac><mo>&#x2248;</mo><mn>0.1515</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3580" style="width: 16.817em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.773em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.361em 1013.73em 3.047em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3581"><span class="mi" id="MathJax-Span-3582" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-3583" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3584" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mo" id="MathJax-Span-3585" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-3586" style="font-family: STIXGeneral; padding-left: 0.33em;">6</span><span class="mo" id="MathJax-Span-3587" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3588" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="msubsup" id="MathJax-Span-3589" style="padding-left: 0.33em;"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px;"><span style="position: absolute; clip: rect(3.422em 1000.42em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-3590" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.354em; left: 0.424em;"><span class="texatom" id="MathJax-Span-3591"><span class="mrow" id="MathJax-Span-3592"><span class="mo" id="MathJax-Span-3593" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-3594" style="font-size: 70.7%; font-family: STIXGeneral;">5.2</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-3595" style="font-family: STIXGeneral; padding-left: 0.237em;">×</span><span class="mfrac" id="MathJax-Span-3596" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 1.314em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3.187em 1001.17em 4.124em -999.998em); top: -4.4em; left: 50%; margin-left: -0.607em;"><span class="msubsup" id="MathJax-Span-3597"><span style="display: inline-block; position: relative; width: 1.173em; height: 0px;"><span style="position: absolute; clip: rect(3.375em 1000.85em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mn" id="MathJax-Span-3598" style="font-size: 70.7%; font-family: STIXGeneral;">5.2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.26em; left: 0.892em;"><span class="mn" id="MathJax-Span-3599" style="font-size: 50%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.375em 1000.52em 4.124em -999.998em); top: -3.604em; left: 50%; margin-left: -0.279em;"><span class="mrow" id="MathJax-Span-3600"><span class="mn" id="MathJax-Span-3601" style="font-size: 70.7%; font-family: STIXGeneral;">6</span><span class="mo" id="MathJax-Span-3602" style="font-size: 70.7%; font-family: STIXGeneral;">!</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1001.31em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.314em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span><span class="mo" id="MathJax-Span-3603" style="font-family: STIXGeneral; padding-left: 0.33em;">≈</span><span class="mn" id="MathJax-Span-3604" style="font-family: STIXGeneral; padding-left: 0.33em;">0.1515</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.511em; border-left: 0px solid; width: 0px; height: 1.889em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>6</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>5.2</mn></mrow></msup><mo>×</mo><mfrac><msup><mn>5.2</mn><mn>6</mn></msup><mrow><mn>6</mn><mo>!</mo></mrow></mfrac><mo>≈</mo><mn>0.1515</mn></math></span></span><script type="math/tex" id="MathJax-Element-261">P(X = 6) = e^{-5.2}\times \frac{5.2^6}{6!} \approx 0.1515</script></span></p>
</blockquote><h3 id="binomial"><a class="anchor hidden-xs" href="#binomial" title="binomial"><span class="octicon octicon-link"></span></a>Binomial</h3><p>Questions on the poisson distribution can include the use of the binomial theorem.</p><p>Example (Following from the above example)</p><blockquote>
<p>What is the probability that exactly <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-262-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3605" style="width: 0.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1000.47em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3606"><span class="mn" id="MathJax-Span-3607" style="font-family: STIXGeneral;">6</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-262">6</script></span> cars pass the checkpoint in at least <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-263-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3608" style="width: 0.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1000.47em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3609"><span class="mn" id="MathJax-Span-3610" style="font-family: STIXGeneral;">3</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-263">3</script></span> or the next <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-264-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3611" style="width: 0.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1000.47em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3612"><span class="mn" id="MathJax-Span-3613" style="font-family: STIXGeneral;">4</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.917em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-264">4</script></span> minutes?<br>
Probability of success = <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-265-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.1515</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3614" style="width: 3.375em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.766em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1002.72em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3615"><span class="mn" id="MathJax-Span-3616" style="font-family: STIXGeneral;">0.1515</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.1515</mn></math></span></span><script type="math/tex" id="MathJax-Element-265">0.1515</script></span>, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-266-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3617" style="width: 2.859em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.344em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1002.3em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3618"><span class="mi" id="MathJax-Span-3619" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-3620" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-3621" style="font-family: STIXGeneral; padding-left: 0.33em;">4</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-266">n = 4</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-267-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&#x2265;</mo><mn>3</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>4</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow></mrow><mo stretchy="false">(</mo><mn>0.1515</mn><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>4</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow></mrow><mo stretchy="false">(</mo><mn>0.1515</mn><msup><mo stretchy="false">)</mo><mn>4</mn></msup></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>&#x2248;</mo><mn>0.0123</mn></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3622" style="width: 22.438em; display: inline-block;"><span style="display: inline-block; position: relative; width: 18.363em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-0.794em 1018.18em 4.593em -999.998em); top: -2.152em; left: 0em;"><span class="mrow" id="MathJax-Span-3623"><span class="mtable" id="MathJax-Span-3624" style="padding-right: 0.19em; padding-left: 0.19em;"><span style="display: inline-block; position: relative; width: 17.988em; height: 0px;"><span style="position: absolute; clip: rect(2.953em 1003.75em 8.059em -999.998em); top: -5.806em; left: 0em;"><span style="display: inline-block; position: relative; width: 3.796em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1003.75em 4.311em -999.998em); top: -5.993em; right: 0em;"><span class="mtd" id="MathJax-Span-3625"><span class="mrow" id="MathJax-Span-3626"><span class="mi" id="MathJax-Span-3627" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-3628" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3629" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mo" id="MathJax-Span-3630" style="font-family: STIXGeneral; padding-left: 0.33em;">≥</span><span class="mn" id="MathJax-Span-3631" style="font-family: STIXGeneral; padding-left: 0.33em;">3</span><span class="mo" id="MathJax-Span-3632" style="font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.843em 1000em 4.124em -999.998em); top: -3.979em; right: 0em;"><span class="mtd" id="MathJax-Span-3650"><span class="mrow" id="MathJax-Span-3651"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.843em 1000em 4.124em -999.998em); top: -1.871em; right: 0em;"><span class="mtd" id="MathJax-Span-3695"><span class="mrow" id="MathJax-Span-3696"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.81em;"></span></span><span style="position: absolute; clip: rect(2.953em 1014.19em 8.105em -999.998em); top: -5.806em; left: 3.796em;"><span style="display: inline-block; position: relative; width: 14.194em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1010.03em 4.311em -999.998em); top: -5.993em; left: 0em;"><span class="mtd" id="MathJax-Span-3633"><span class="mrow" id="MathJax-Span-3634"><span class="mi" id="MathJax-Span-3635"></span><span class="mo" id="MathJax-Span-3636" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mi" id="MathJax-Span-3637" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.33em;">P</span><span class="mo" id="MathJax-Span-3638" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3639" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mo" id="MathJax-Span-3640" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-3641" style="font-family: STIXGeneral; padding-left: 0.33em;">3</span><span class="mo" id="MathJax-Span-3642" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3643" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="mi" id="MathJax-Span-3644" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.237em;">P</span><span class="mo" id="MathJax-Span-3645" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3646" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mo" id="MathJax-Span-3647" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-3648" style="font-family: STIXGeneral; padding-left: 0.33em;">4</span><span class="mo" id="MathJax-Span-3649" style="font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.344em 1014.19em 5.108em -999.998em); top: -3.979em; left: 0em;"><span class="mtd" id="MathJax-Span-3652"><span class="mrow" id="MathJax-Span-3653"><span class="mi" id="MathJax-Span-3654"></span><span class="mo" id="MathJax-Span-3655" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="texatom" id="MathJax-Span-3656" style="padding-left: 0.33em;"><span class="mrow" id="MathJax-Span-3657"><span class="mrow" id="MathJax-Span-3658"><span class="TeXmathchoice" id="MathJax-Span-3659"><span class="texatom" id="MathJax-Span-3660"><span class="mrow" id="MathJax-Span-3661"><span class="mo" id="MathJax-Span-3662" style="vertical-align: -0.607em;"><span style="font-family: STIXSizeThreeSym;">(</span></span></span></span></span><span class="mfrac" id="MathJax-Span-3663"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.47em 4.124em -999.998em); top: -4.635em; left: 50%; margin-left: -0.232em;"><span class="mn" id="MathJax-Span-3664" style="font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1000.42em 4.124em -999.998em); top: -3.276em; left: 50%; margin-left: -0.232em;"><span class="mn" id="MathJax-Span-3665" style="font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="TeXmathchoice" id="MathJax-Span-3666"><span class="texatom" id="MathJax-Span-3667"><span class="mrow" id="MathJax-Span-3668"><span class="mo" id="MathJax-Span-3669" style="vertical-align: -0.607em;"><span style="font-family: STIXSizeThreeSym;">)</span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-3670" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-3671" style="font-family: STIXGeneral;">0.1515</span><span class="msubsup" id="MathJax-Span-3672"><span style="display: inline-block; position: relative; width: 0.752em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.28em 4.311em -999.998em); top: -3.979em; left: 0em;"><span class="mo" id="MathJax-Span-3673" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.4em; left: 0.33em;"><span class="mn" id="MathJax-Span-3674" style="font-size: 70.7%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-3675" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="texatom" id="MathJax-Span-3676" style="padding-left: 0.237em;"><span class="mrow" id="MathJax-Span-3677"><span class="mrow" id="MathJax-Span-3678"><span class="TeXmathchoice" id="MathJax-Span-3679"><span class="texatom" id="MathJax-Span-3680"><span class="mrow" id="MathJax-Span-3681"><span class="mo" id="MathJax-Span-3682" style="vertical-align: -0.607em;"><span style="font-family: STIXSizeThreeSym;">(</span></span></span></span></span><span class="mfrac" id="MathJax-Span-3683"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.47em 4.124em -999.998em); top: -4.635em; left: 50%; margin-left: -0.232em;"><span class="mn" id="MathJax-Span-3684" style="font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1000.47em 4.124em -999.998em); top: -3.276em; left: 50%; margin-left: -0.232em;"><span class="mn" id="MathJax-Span-3685" style="font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="TeXmathchoice" id="MathJax-Span-3686"><span class="texatom" id="MathJax-Span-3687"><span class="mrow" id="MathJax-Span-3688"><span class="mo" id="MathJax-Span-3689" style="vertical-align: -0.607em;"><span style="font-family: STIXSizeThreeSym;">)</span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-3690" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-3691" style="font-family: STIXGeneral;">0.1515</span><span class="msubsup" id="MathJax-Span-3692"><span style="display: inline-block; position: relative; width: 0.752em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.28em 4.311em -999.998em); top: -3.979em; left: 0em;"><span class="mo" id="MathJax-Span-3693" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.4em; left: 0.33em;"><span class="mn" id="MathJax-Span-3694" style="font-size: 70.7%; font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1004.03em 4.124em -999.998em); top: -1.871em; left: 0em;"><span class="mtd" id="MathJax-Span-3697"><span class="mrow" id="MathJax-Span-3698"><span class="mi" id="MathJax-Span-3699"></span><span class="mo" id="MathJax-Span-3700" style="font-family: STIXGeneral; padding-left: 0.33em;">≈</span><span class="mn" id="MathJax-Span-3701" style="font-family: STIXGeneral; padding-left: 0.33em;">0.0123</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.81em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.157em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.854em; border-left: 0px solid; width: 0px; height: 6.403em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mn>3</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>4</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow></mrow><mo stretchy="false">(</mo><mn>0.1515</mn><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>4</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow></mrow><mo stretchy="false">(</mo><mn>0.1515</mn><msup><mo stretchy="false">)</mo><mn>4</mn></msup></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>≈</mo><mn>0.0123</mn></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-267">\begin{align} P(X \geq 3) &= P(X = 3) + P(X = 4) \\ &= {4 \choose 3}(0.1515)^3 + {4 \choose 4}(0.1515)^4 \\ &\approx 0.0123 \end{align}</script></span></p>
</blockquote><h3 id="mean-and-variance-of-a-poisson-distribution"><a class="anchor hidden-xs" href="#mean-and-variance-of-a-poisson-distribution" title="mean-and-variance-of-a-poisson-distribution"><span class="octicon octicon-link"></span></a>Mean and variance of a Poisson distribution</h3><p>Mean = Variance = <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-153-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2139" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2140"><span class="mi" id="MathJax-Span-2141" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-153">\lambda</script></span></p><h2 id="continuous-random-variables"><a class="anchor hidden-xs" href="#continuous-random-variables" title="continuous-random-variables"><span class="octicon octicon-link"></span></a>Continuous random variables</h2><p><strong>Continuous random variable</strong> A variable which can take an infinite number of possible values</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-154-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace linebreak="newline" /><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&lt;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&#x2264;</mo><mi>t</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2142" style="width: 100%; display: inline-block; min-width: 10.197em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(3.179em 1008.3em 5.689em -999.997em); top: -3.993em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-2143"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1005.53em 4.306em -999.997em); top: -3.993em; left: 50%; margin-left: -2.764em;"><span class="mi" id="MathJax-Span-2144" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2145" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2146" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2147" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-2148" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span><span class="mo" id="MathJax-Span-2149" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2150" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-2151" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1008.3em 4.306em -999.997em); top: -2.661em; left: 50%; margin-left: -4.198em;"><span class="mspace" id="MathJax-Span-2152" style="height: 0em; vertical-align: 0em; width: 0em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-2153" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2154" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2155" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2156" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mi" id="MathJax-Span-2157" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2158" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2159" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2160" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">P</span><span class="mo" id="MathJax-Span-2161" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2162" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2163" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mi" id="MathJax-Span-2164" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2165" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.934em; border-left: 0px solid; width: 0px; height: 2.816em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace linebreak="newline"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo><</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-154">P(X = 0) = 0 \\ P(X < t) = P(X \leq t)</script></span></p><h3 id="probability-density-functions-fx"><a class="anchor hidden-xs" href="#probability-density-functions-fx" title="probability-density-functions-fx"><span class="octicon octicon-link"></span></a>Probability density functions <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-155-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2166" style="width: 1.928em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.559em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.682em 1001.52em 2.83em -999.998em); top: -2.498em; left: 0em;"><span class="mrow" id="MathJax-Span-2167"><span class="mi" id="MathJax-Span-2168" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.166em;"></span></span><span class="mo" id="MathJax-Span-2169" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2170" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-2171" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.502em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.297em; border-left: 0px solid; width: 0px; height: 1.203em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-155">f(x)</script></span></h3><ul>
<li>The area under the graph of a probability density function represents the probability</li>
<li>The total area under the graph must always equal <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-156-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2172" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2173"><span class="mn" id="MathJax-Span-2174" style="font-family: STIXGeneral;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-156">1</script></span></li>
</ul><h3 id="cumulative-distribution-function-fx"><a class="anchor hidden-xs" href="#cumulative-distribution-function-fx" title="cumulative-distribution-function-fx"><span class="octicon octicon-link"></span></a>Cumulative distribution function <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-157-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2175" style="width: 2.174em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.764em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.723em 1001.72em 2.789em -999.998em); top: -2.498em; left: 0em;"><span class="mrow" id="MathJax-Span-2176"><span class="mi" id="MathJax-Span-2177" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.043em;"></span></span><span class="mo" id="MathJax-Span-2178" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2179" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-2180" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.502em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.153em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-157">F(x)</script></span></h3><ul>
<li>Is the integral of the probability density function <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-158-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2181" style="width: 1.898em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.49em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2182"><span class="mi" id="MathJax-Span-2183" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-2184" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2185" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2186" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-158">f(x)</script></span></li>
<li>Gives the probability that the values is less than <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-159-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2187" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2188"><span class="mi" id="MathJax-Span-2189" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-159">x</script></span>. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-160-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&lt;</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&#x2264;</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2190" style="width: 10.556em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.66em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1008.61em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2191"><span class="mi" id="MathJax-Span-2192" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2193" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2194" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2195" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mi" id="MathJax-Span-2196" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2197" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2198" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2199" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">P</span><span class="mo" id="MathJax-Span-2200" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2201" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2202" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mi" id="MathJax-Span-2203" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2204" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo><</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-160">P(X < x) = P(X \leq x)</script></span></li>
<li>Useful when finding medians and percentiles. e.g. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-161-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0.5</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2205" style="width: 5.279em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.306em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1004.25em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2206"><span class="mi" id="MathJax-Span-2207" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2208" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2209" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2210" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2211" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-2212" style="font-family: STIXGeneral; padding-left: 0.31em;">0.5</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0.5</mn></math></span></span><script type="math/tex" id="MathJax-Element-161">F(x) = 0.5</script></span> gives the median</li>
</ul><p>Example</p><blockquote>
<p>Find <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-268-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3702" style="width: 2.204em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.782em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1001.74em 2.859em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3703"><span class="mi" id="MathJax-Span-3704" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mo" id="MathJax-Span-3705" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3706" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3707" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.283em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-268">F(x)</script></span> for the following probability density function<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-269-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>18</mn></mfrac></mtd><mtd><mspace width="1em" /><mn>0</mn><mo>&#x2264;</mo><mi>x</mi><mo>&#x2264;</mo><mn>3</mn></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>5</mn><mo>&#x2212;</mo><mi>x</mi><mo stretchy="false">)</mo></mtd><mtd><mspace width="1em" /><mn>3</mn><mo>&#x2264;</mo><mi>x</mi><mo>&#x2264;</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mspace width="1em" /><mi>o</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>w</mi><mi>i</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3708" style="width: 16.817em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.773em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.204em 1013.77em 7.028em -999.998em); top: -4.869em; left: 0em;"><span class="mrow" id="MathJax-Span-3709"><span class="mi" id="MathJax-Span-3710" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.143em;"></span></span><span class="mo" id="MathJax-Span-3711" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3712" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3713" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3714" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mrow" id="MathJax-Span-3715" style="padding-left: 0.33em;"><span class="mo" id="MathJax-Span-3716" style="vertical-align: 2.532em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: STIXSizeOneSym; top: -3.276em; left: 0em;">⎧<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXSizeOneSym; top: 0.237em; left: 0em;">⎩<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXSizeOneSym; top: -1.496em; left: 0em;">⎨<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXSizeOneSym; position: absolute; top: -2.386em; left: 0em;">⎪<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXSizeOneSym; position: absolute; top: -0.607em; left: 0em;">⎪<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mtable" id="MathJax-Span-3717" style="padding-right: 0.19em; padding-left: 0.19em;"><span style="display: inline-block; position: relative; width: 9.885em; height: 0px;"><span style="position: absolute; clip: rect(2.672em 1003.42em 7.309em -999.998em); top: -5.337em; left: 0em;"><span style="display: inline-block; position: relative; width: 3.468em; height: 0px;"><span style="position: absolute; clip: rect(2.859em 1001.08em 4.499em -999.998em); top: -5.478em; left: 0em;"><span class="mtd" id="MathJax-Span-3718"><span class="mrow" id="MathJax-Span-3719"><span class="mfrac" id="MathJax-Span-3720"><span style="display: inline-block; position: relative; width: 0.845em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3.234em 1000.66em 4.124em -999.998em); top: -4.4em; left: 50%; margin-left: -0.326em;"><span class="msubsup" id="MathJax-Span-3721"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; clip: rect(3.515em 1000.33em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-3722" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.213em; left: 0.377em;"><span class="mn" id="MathJax-Span-3723" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.375em 1000.66em 4.124em -999.998em); top: -3.604em; left: 50%; margin-left: -0.372em;"><span class="mn" id="MathJax-Span-3724" style="font-size: 70.7%; font-family: STIXGeneral;">18</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1000.85em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.845em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.953em 1003.42em 4.499em -999.998em); top: -3.792em; left: 0em;"><span class="mtd" id="MathJax-Span-3733"><span class="mrow" id="MathJax-Span-3734"><span class="mfrac" id="MathJax-Span-3735"><span style="display: inline-block; position: relative; width: 0.471em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3.375em 1000.28em 4.124em -999.998em); top: -4.354em; left: 50%; margin-left: -0.185em;"><span class="mn" id="MathJax-Span-3736" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.375em 1000.33em 4.124em -999.998em); top: -3.604em; left: 50%; margin-left: -0.185em;"><span class="mn" id="MathJax-Span-3737" style="font-size: 70.7%; font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1000.47em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.471em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span><span class="mo" id="MathJax-Span-3738" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-3739" style="font-family: STIXGeneral;">5</span><span class="mo" id="MathJax-Span-3740" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="mi" id="MathJax-Span-3741" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.237em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3742" style="font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1000.47em 4.124em -999.998em); top: -2.152em; left: 0em;"><span class="mtd" id="MathJax-Span-3751"><span class="mrow" id="MathJax-Span-3752"><span class="mn" id="MathJax-Span-3753" style="font-family: STIXGeneral;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.342em;"></span></span><span style="position: absolute; clip: rect(2.672em 1005.2em 6.981em -999.998em); top: -5.009em; left: 4.593em;"><span style="display: inline-block; position: relative; width: 5.248em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1005.2em 4.218em -999.998em); top: -5.478em; left: 0em;"><span class="mtd" id="MathJax-Span-3725"><span class="mrow" id="MathJax-Span-3726"><span class="mspace" id="MathJax-Span-3727" style="height: 0em; vertical-align: 0em; width: 1.126em; display: inline-block; overflow: hidden;"></span><span class="mn" id="MathJax-Span-3728" style="font-family: STIXGeneral;">0</span><span class="mo" id="MathJax-Span-3729" style="font-family: STIXGeneral; padding-left: 0.33em;">≤</span><span class="mi" id="MathJax-Span-3730" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.33em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3731" style="font-family: STIXGeneral; padding-left: 0.33em;">≤</span><span class="mn" id="MathJax-Span-3732" style="font-family: STIXGeneral; padding-left: 0.33em;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.141em 1005.2em 4.218em -999.998em); top: -3.792em; left: 0em;"><span class="mtd" id="MathJax-Span-3743"><span class="mrow" id="MathJax-Span-3744"><span class="mspace" id="MathJax-Span-3745" style="height: 0em; vertical-align: 0em; width: 1.126em; display: inline-block; overflow: hidden;"></span><span class="mn" id="MathJax-Span-3746" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-3747" style="font-family: STIXGeneral; padding-left: 0.33em;">≤</span><span class="mi" id="MathJax-Span-3748" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.33em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3749" style="font-family: STIXGeneral; padding-left: 0.33em;">≤</span><span class="mn" id="MathJax-Span-3750" style="font-family: STIXGeneral; padding-left: 0.33em;">5</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.141em 1005.01em 4.124em -999.998em); top: -2.152em; left: 0em;"><span class="mtd" id="MathJax-Span-3754"><span class="mrow" id="MathJax-Span-3755"><span class="mspace" id="MathJax-Span-3756" style="height: 0em; vertical-align: 0em; width: 1.126em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-3757" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-3758" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-3759" style="font-family: STIXGeneral; font-style: italic;">h</span><span class="mi" id="MathJax-Span-3760" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mi" id="MathJax-Span-3761" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-3762" style="font-family: STIXGeneral; font-style: italic;">w</span><span class="mi" id="MathJax-Span-3763" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-3764" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-3765" style="font-family: STIXGeneral; font-style: italic;">e</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.014em;"></span></span></span></span><span class="mo" id="MathJax-Span-3766"></span></span></span><span style="display: inline-block; width: 0px; height: 4.874em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.511em; border-left: 0px solid; width: 0px; height: 5.66em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>18</mn></mfrac></mtd><mtd><mspace width="1em"></mspace><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>3</mn></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>5</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mtd><mtd><mspace width="1em"></mspace><mn>3</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mspace width="1em"></mspace><mi>o</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>w</mi><mi>i</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-269">f(x) = \left\{
\begin{array}{lll}
\frac{x^2}{18} & \quad 0 \leq x \leq 3 \\
\frac{1}{4}(5-x) & \quad 3 \leq x \leq 5 \\
0 & \quad otherwise \\
\end{array}
\right.</script></span><br>
The function <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-270-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3767" style="width: 1.923em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.548em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1001.5em 2.859em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3768"><span class="mi" id="MathJax-Span-3769" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.143em;"></span></span><span class="mo" id="MathJax-Span-3770" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3771" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3772" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.283em; border-left: 0px solid; width: 0px; height: 1.203em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-270">f(x)</script></span> must be integrated in sections<br>
The first section is a quadratic. If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-271-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>c</mi><mo>&lt;</mo><mn>3</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3773" style="width: 5.061em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.124em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1004.08em 2.719em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3774"><span class="mn" id="MathJax-Span-3775" style="font-family: STIXGeneral;">0</span><span class="mo" id="MathJax-Span-3776" style="font-family: STIXGeneral; padding-left: 0.33em;"><</span><span class="mi" id="MathJax-Span-3777" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.33em;">c</span><span class="mo" id="MathJax-Span-3778" style="font-family: STIXGeneral; padding-left: 0.33em;"><</span><span class="mn" id="MathJax-Span-3779" style="font-family: STIXGeneral; padding-left: 0.33em;">3</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.111em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>c</mi><mo><</mo><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-271">0 < c < 3</script></span> then<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-272-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&lt;</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>&#x222B;</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>c</mi></mrow></msubsup><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>18</mn></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><msup><mi>c</mi><mn>3</mn></msup><mn>54</mn></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3780" style="width: 14.663em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.993em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.892em 1011.99em 3.609em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3781"><span class="mi" id="MathJax-Span-3782" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-3783" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3784" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mo" id="MathJax-Span-3785" style="font-family: STIXGeneral; padding-left: 0.33em;"><</span><span class="mi" id="MathJax-Span-3786" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.33em;">c</span><span class="mo" id="MathJax-Span-3787" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3788" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="msubsup" id="MathJax-Span-3789" style="padding-left: 0.33em;"><span style="display: inline-block; position: relative; width: 1.595em; height: 0px;"><span style="position: absolute; clip: rect(2.438em 1001.03em 5.014em -999.998em); top: -3.979em; left: 0em;"><span class="mo" id="MathJax-Span-3790" style="font-family: STIXIntegralsD; vertical-align: -0.607em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.471em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.515em 1000.38em 4.124em -999.998em); top: -5.103em; left: 1.22em;"><span class="texatom" id="MathJax-Span-3791"><span class="mrow" id="MathJax-Span-3792"><span class="mi" id="MathJax-Span-3793" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">c</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.375em 1000.42em 4.124em -999.998em); top: -3.042em; left: 0.564em;"><span class="texatom" id="MathJax-Span-3794"><span class="mrow" id="MathJax-Span-3795"><span class="mn" id="MathJax-Span-3796" style="font-size: 70.7%; font-family: STIXGeneral;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-3797" style="padding-left: 0.19em;"><span style="display: inline-block; position: relative; width: 1.126em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3em 1000.94em 4.124em -999.998em); top: -4.635em; left: 50%; margin-left: -0.466em;"><span class="msubsup" id="MathJax-Span-3798"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px;"><span style="position: absolute; clip: rect(3.422em 1000.47em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-3799" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.354em; left: 0.518em;"><span class="mn" id="MathJax-Span-3800" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1000.94em 4.124em -999.998em); top: -3.276em; left: 50%; margin-left: -0.513em;"><span class="mn" id="MathJax-Span-3801" style="font-family: STIXGeneral;">18</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1001.13em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.126em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span><span class="mi" id="MathJax-Span-3802" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mi" id="MathJax-Span-3803" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3804" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mfrac" id="MathJax-Span-3805" style="padding-left: 0.33em;"><span style="display: inline-block; position: relative; width: 1.126em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3em 1000.89em 4.124em -999.998em); top: -4.635em; left: 50%; margin-left: -0.419em;"><span class="msubsup" id="MathJax-Span-3806"><span style="display: inline-block; position: relative; width: 0.892em; height: 0px;"><span style="position: absolute; clip: rect(3.422em 1000.42em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-3807" style="font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.354em; left: 0.424em;"><span class="mn" id="MathJax-Span-3808" style="font-size: 70.7%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.141em 1000.99em 4.124em -999.998em); top: -3.276em; left: 50%; margin-left: -0.513em;"><span class="mn" id="MathJax-Span-3809" style="font-family: STIXGeneral;">54</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1001.13em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.126em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.197em; border-left: 0px solid; width: 0px; height: 3.089em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo><</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>c</mi></mrow></msubsup><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>18</mn></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><msup><mi>c</mi><mn>3</mn></msup><mn>54</mn></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-272">P(X < c) = \int_{0}^{c} \frac{x^2}{18} dx =\frac{c^3}{54} </script></span><br>
Using the above formula, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-273-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&lt;</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.5</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3810" style="width: 7.778em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.372em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1006.33em 2.859em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3811"><span class="mi" id="MathJax-Span-3812" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-3813" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3814" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mo" id="MathJax-Span-3815" style="font-family: STIXGeneral; padding-left: 0.33em;"><</span><span class="mn" id="MathJax-Span-3816" style="font-family: STIXGeneral; padding-left: 0.33em;">3</span><span class="mo" id="MathJax-Span-3817" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3818" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-3819" style="font-family: STIXGeneral; padding-left: 0.33em;">0.5</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.283em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo><</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.5</mn></math></span></span><script type="math/tex" id="MathJax-Element-273">P(X < 3) = 0.5</script></span><br>
The second section is linear. If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-274-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&lt;</mo><mi>c</mi><mo>&lt;</mo><mn>5</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3820" style="width: 5.061em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.124em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1004.08em 2.719em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3821"><span class="mn" id="MathJax-Span-3822" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-3823" style="font-family: STIXGeneral; padding-left: 0.33em;"><</span><span class="mi" id="MathJax-Span-3824" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.33em;">c</span><span class="mo" id="MathJax-Span-3825" style="font-family: STIXGeneral; padding-left: 0.33em;"><</span><span class="mn" id="MathJax-Span-3826" style="font-family: STIXGeneral; padding-left: 0.33em;">5</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.111em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo><</mo><mi>c</mi><mo><</mo><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-274">3 < c < 5</script></span> then<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-275-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&lt;</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><msubsup><mo>&#x222B;</mo><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>c</mi></mrow></msubsup><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>5</mn><mo>&#x2212;</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mo stretchy="false">(</mo><mn>10</mn><mi>c</mi><mo>&#x2212;</mo><msup><mi>c</mi><mn>2</mn></msup><mo>&#x2212;</mo><mn>17</mn><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3827" style="width: 26.934em; display: inline-block;"><span style="display: inline-block; position: relative; width: 22.063em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.986em 1022.02em 3.609em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3828"><span class="mi" id="MathJax-Span-3829" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-3830" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3831" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mo" id="MathJax-Span-3832" style="font-family: STIXGeneral; padding-left: 0.33em;"><</span><span class="mi" id="MathJax-Span-3833" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.33em;">c</span><span class="mo" id="MathJax-Span-3834" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3835" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mfrac" id="MathJax-Span-3836" style="padding-left: 0.33em;"><span style="display: inline-block; position: relative; width: 0.611em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3.187em 1000.38em 4.124em -999.998em); top: -4.635em; left: 50%; margin-left: -0.232em;"><span class="mn" id="MathJax-Span-3837" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1000.47em 4.124em -999.998em); top: -3.276em; left: 50%; margin-left: -0.232em;"><span class="mn" id="MathJax-Span-3838" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1000.61em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.611em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span><span class="mo" id="MathJax-Span-3839" style="font-family: STIXGeneral; padding-left: 0.237em;">+</span><span class="msubsup" id="MathJax-Span-3840" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 1.595em; height: 0px;"><span style="position: absolute; clip: rect(2.438em 1001.03em 5.014em -999.998em); top: -3.979em; left: 0em;"><span class="mo" id="MathJax-Span-3841" style="font-family: STIXIntegralsD; vertical-align: -0.607em;">∫<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.471em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.515em 1000.38em 4.124em -999.998em); top: -5.103em; left: 1.22em;"><span class="texatom" id="MathJax-Span-3842"><span class="mrow" id="MathJax-Span-3843"><span class="mi" id="MathJax-Span-3844" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">c</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.375em 1000.42em 4.124em -999.998em); top: -3.042em; left: 0.564em;"><span class="texatom" id="MathJax-Span-3845"><span class="mrow" id="MathJax-Span-3846"><span class="mn" id="MathJax-Span-3847" style="font-size: 70.7%; font-family: STIXGeneral;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-3848" style="padding-left: 0.19em;"><span style="display: inline-block; position: relative; width: 0.611em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3.187em 1000.38em 4.124em -999.998em); top: -4.635em; left: 50%; margin-left: -0.232em;"><span class="mn" id="MathJax-Span-3849" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1000.47em 4.124em -999.998em); top: -3.276em; left: 50%; margin-left: -0.232em;"><span class="mn" id="MathJax-Span-3850" style="font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1000.61em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.611em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span><span class="mo" id="MathJax-Span-3851" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-3852" style="font-family: STIXGeneral;">5</span><span class="mo" id="MathJax-Span-3853" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="mi" id="MathJax-Span-3854" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.237em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3855" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-3856" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mi" id="MathJax-Span-3857" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3858" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mfrac" id="MathJax-Span-3859" style="padding-left: 0.33em;"><span style="display: inline-block; position: relative; width: 0.611em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3.187em 1000.38em 4.124em -999.998em); top: -4.635em; left: 50%; margin-left: -0.232em;"><span class="mn" id="MathJax-Span-3860" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1000.47em 4.124em -999.998em); top: -3.276em; left: 50%; margin-left: -0.232em;"><span class="mn" id="MathJax-Span-3861" style="font-family: STIXGeneral;">8</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1000.61em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.611em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span><span class="mo" id="MathJax-Span-3862" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-3863" style="font-family: STIXGeneral;">10</span><span class="mi" id="MathJax-Span-3864" style="font-family: STIXGeneral; font-style: italic;">c</span><span class="mo" id="MathJax-Span-3865" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="msubsup" id="MathJax-Span-3866" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 0.892em; height: 0px;"><span style="position: absolute; clip: rect(3.422em 1000.42em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-3867" style="font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.4em; left: 0.424em;"><span class="mn" id="MathJax-Span-3868" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-3869" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="mn" id="MathJax-Span-3870" style="font-family: STIXGeneral; padding-left: 0.237em;">17</span><span class="mo" id="MathJax-Span-3871" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.197em; border-left: 0px solid; width: 0px; height: 2.974em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo><</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><msubsup><mo>∫</mo><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>c</mi></mrow></msubsup><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>5</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mo stretchy="false">(</mo><mn>10</mn><mi>c</mi><mo>−</mo><msup><mi>c</mi><mn>2</mn></msup><mo>−</mo><mn>17</mn><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-275">P(X < c) = \frac{1}{2} + \int_{3}^{c} \frac{1}{4}(5-x) dx = \frac{1}{8}(10c - c^2 -17)</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-276-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3872" style="width: 2.204em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.782em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1001.74em 2.859em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3873"><span class="mi" id="MathJax-Span-3874" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mo" id="MathJax-Span-3875" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3876" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3877" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.283em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-276">F(x)</script></span> is then given by the piecewise function<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-277-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>0</mn></mtd><mtd><mspace width="1em" /><mi>x</mi><mo>&lt;</mo><mn>0</mn></mtd></mtr><mtr><mtd><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>54</mn></mfrac></mtd><mtd><mspace width="1em" /><mn>0</mn><mo>&#x2264;</mo><mi>x</mi><mo>&#x2264;</mo><mn>3</mn></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>8</mn></mfrac><mo stretchy="false">(</mo><mn>10</mn><mi>x</mi><mo>&#x2212;</mo><msup><mi>x</mi><mn>2</mn></msup><mo>&#x2212;</mo><mn>17</mn><mo stretchy="false">)</mo></mtd><mtd><mspace width="1em" /><mn>3</mn><mo>&#x2264;</mo><mi>x</mi><mo>&#x2264;</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mspace width="1em" /><mi>x</mi><mo>&#x2265;</mo><mn>5</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3878" style="width: 21.173em; display: inline-block;"><span style="display: inline-block; position: relative; width: 17.333em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.204em 1017.33em 8.48em -999.998em); top: -5.571em; left: 0em;"><span class="mrow" id="MathJax-Span-3879"><span class="mi" id="MathJax-Span-3880" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.143em;"></span></span><span class="mo" id="MathJax-Span-3881" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3882" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3883" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-3884" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mrow" id="MathJax-Span-3885" style="padding-left: 0.33em;"><span class="mo" id="MathJax-Span-3886" style="vertical-align: 3.234em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: STIXSizeOneSym; top: -3.276em; left: 0em;">⎧<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXSizeOneSym; top: 1.735em; left: 0em;">⎩<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXSizeOneSym; top: -0.794em; left: 0em;">⎨<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXSizeOneSym; position: absolute; top: -2.433em; left: 0em;">⎪<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXSizeOneSym; position: absolute; top: -1.59em; left: 0em;">⎪<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXSizeOneSym; position: absolute; top: 0.049em; left: 0em;">⎪<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXSizeOneSym; position: absolute; top: 0.892em; left: 0em;">⎪<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mtable" id="MathJax-Span-3887" style="padding-right: 0.19em; padding-left: 0.19em;"><span style="display: inline-block; position: relative; width: 13.445em; height: 0px;"><span style="position: absolute; clip: rect(3.422em 1007.03em 9.417em -999.998em); top: -6.696em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.075em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.47em 4.124em -999.998em); top: -6.415em; left: 0em;"><span class="mtd" id="MathJax-Span-3888"><span class="mrow" id="MathJax-Span-3889"><span class="mn" id="MathJax-Span-3890" style="font-family: STIXGeneral;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.859em 1001.08em 4.499em -999.998em); top: -4.775em; left: 0em;"><span class="mtd" id="MathJax-Span-3897"><span class="mrow" id="MathJax-Span-3898"><span class="mfrac" id="MathJax-Span-3899"><span style="display: inline-block; position: relative; width: 0.845em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3.234em 1000.66em 4.124em -999.998em); top: -4.4em; left: 50%; margin-left: -0.326em;"><span class="msubsup" id="MathJax-Span-3900"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; clip: rect(3.515em 1000.33em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-3901" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.213em; left: 0.377em;"><span class="mn" id="MathJax-Span-3902" style="font-size: 50%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.375em 1000.71em 4.124em -999.998em); top: -3.604em; left: 50%; margin-left: -0.372em;"><span class="mn" id="MathJax-Span-3903" style="font-size: 70.7%; font-family: STIXGeneral;">54</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1000.85em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.845em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.953em 1007.03em 4.499em -999.998em); top: -3.042em; left: 0em;"><span class="mtd" id="MathJax-Span-3912"><span class="mrow" id="MathJax-Span-3913"><span class="mfrac" id="MathJax-Span-3914"><span style="display: inline-block; position: relative; width: 0.471em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3.375em 1000.28em 4.124em -999.998em); top: -4.354em; left: 50%; margin-left: -0.185em;"><span class="mn" id="MathJax-Span-3915" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.375em 1000.33em 4.124em -999.998em); top: -3.604em; left: 50%; margin-left: -0.185em;"><span class="mn" id="MathJax-Span-3916" style="font-size: 70.7%; font-family: STIXGeneral;">8</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1000.47em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.471em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span><span class="mo" id="MathJax-Span-3917" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-3918" style="font-family: STIXGeneral;">10</span><span class="mi" id="MathJax-Span-3919" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3920" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="msubsup" id="MathJax-Span-3921" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px;"><span style="position: absolute; clip: rect(3.422em 1000.47em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-3922" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.354em; left: 0.518em;"><span class="mn" id="MathJax-Span-3923" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-3924" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="mn" id="MathJax-Span-3925" style="font-family: STIXGeneral; padding-left: 0.237em;">17</span><span class="mo" id="MathJax-Span-3926" style="font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1000.38em 4.124em -999.998em); top: -1.403em; left: 0em;"><span class="mtd" id="MathJax-Span-3935"><span class="mrow" id="MathJax-Span-3936"><span class="mn" id="MathJax-Span-3937" style="font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.7em;"></span></span><span style="position: absolute; clip: rect(3.515em 1005.2em 9.604em -999.998em); top: -6.789em; left: 8.199em;"><span style="display: inline-block; position: relative; width: 5.248em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1003.38em 4.171em -999.998em); top: -6.415em; left: 0em;"><span class="mtd" id="MathJax-Span-3891"><span class="mrow" id="MathJax-Span-3892"><span class="mspace" id="MathJax-Span-3893" style="height: 0em; vertical-align: 0em; width: 1.126em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-3894" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3895" style="font-family: STIXGeneral; padding-left: 0.33em;"><</span><span class="mn" id="MathJax-Span-3896" style="font-family: STIXGeneral; padding-left: 0.33em;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1005.2em 4.218em -999.998em); top: -4.775em; left: 0em;"><span class="mtd" id="MathJax-Span-3904"><span class="mrow" id="MathJax-Span-3905"><span class="mspace" id="MathJax-Span-3906" style="height: 0em; vertical-align: 0em; width: 1.126em; display: inline-block; overflow: hidden;"></span><span class="mn" id="MathJax-Span-3907" style="font-family: STIXGeneral;">0</span><span class="mo" id="MathJax-Span-3908" style="font-family: STIXGeneral; padding-left: 0.33em;">≤</span><span class="mi" id="MathJax-Span-3909" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.33em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3910" style="font-family: STIXGeneral; padding-left: 0.33em;">≤</span><span class="mn" id="MathJax-Span-3911" style="font-family: STIXGeneral; padding-left: 0.33em;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.141em 1005.2em 4.218em -999.998em); top: -3.042em; left: 0em;"><span class="mtd" id="MathJax-Span-3927"><span class="mrow" id="MathJax-Span-3928"><span class="mspace" id="MathJax-Span-3929" style="height: 0em; vertical-align: 0em; width: 1.126em; display: inline-block; overflow: hidden;"></span><span class="mn" id="MathJax-Span-3930" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-3931" style="font-family: STIXGeneral; padding-left: 0.33em;">≤</span><span class="mi" id="MathJax-Span-3932" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.33em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3933" style="font-family: STIXGeneral; padding-left: 0.33em;">≤</span><span class="mn" id="MathJax-Span-3934" style="font-family: STIXGeneral; padding-left: 0.33em;">5</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.141em 1003.38em 4.218em -999.998em); top: -1.403em; left: 0em;"><span class="mtd" id="MathJax-Span-3938"><span class="mrow" id="MathJax-Span-3939"><span class="mspace" id="MathJax-Span-3940" style="height: 0em; vertical-align: 0em; width: 1.126em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-3941" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3942" style="font-family: STIXGeneral; padding-left: 0.33em;">≥</span><span class="mn" id="MathJax-Span-3943" style="font-family: STIXGeneral; padding-left: 0.33em;">5</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.794em;"></span></span></span></span><span class="mo" id="MathJax-Span-3944"></span></span></span><span style="display: inline-block; width: 0px; height: 5.576em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -3.426em; border-left: 0px solid; width: 0px; height: 7.431em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>0</mn></mtd><mtd><mspace width="1em"></mspace><mi>x</mi><mo><</mo><mn>0</mn></mtd></mtr><mtr><mtd><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>54</mn></mfrac></mtd><mtd><mspace width="1em"></mspace><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>3</mn></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>8</mn></mfrac><mo stretchy="false">(</mo><mn>10</mn><mi>x</mi><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>17</mn><mo stretchy="false">)</mo></mtd><mtd><mspace width="1em"></mspace><mn>3</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mspace width="1em"></mspace><mi>x</mi><mo>≥</mo><mn>5</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-277">f(x) = \left\{
\begin{array}{llll}
0 & \quad x < 0 \\
\frac{x^3}{54} & \quad 0 \leq x \leq 3 \\
\frac{1}{8}(10x - x^2 -17) & \quad 3 \leq x \leq 5 \\
1& \quad x \geq 5 \\
\end{array}
\right.</script></span></p>
</blockquote><h3 id="rectangular--continuous-uniform-distribution"><a class="anchor hidden-xs" href="#rectangular--continuous-uniform-distribution" title="rectangular--continuous-uniform-distribution"><span class="octicon octicon-link"></span></a>Rectangular / Continuous uniform distribution</h3><p>A rectangular distribution is given by</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-172-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mfrac><mn>1</mn><mrow><mi>b</mi><mo>&#x2212;</mo><mi>a</mi></mrow></mfrac></mtd><mtd><mspace width="1em" /><mi>a</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mspace width="1em" /><mi>o</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>w</mi><mi>i</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2456" style="width: 14.757em; display: inline-block;"><span style="display: inline-block; position: relative; width: 12.093em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.205em 1012.09em 5.587em -999.997em); top: -4.147em; left: 0em;"><span class="mrow" id="MathJax-Span-2457"><span class="mi" id="MathJax-Span-2458" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-2459" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2460" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2461" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2462" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mrow" id="MathJax-Span-2463" style="padding-left: 0.31em;"><span class="mo" id="MathJax-Span-2464" style="vertical-align: -0.766em;"><span style="font-family: STIXSizeFourSym;">{</span></span><span class="mtable" id="MathJax-Span-2465" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 7.841em; height: 0px;"><span style="position: absolute; clip: rect(2.205em 1001.54em 5.126em -999.997em); top: -3.993em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px;"><span style="position: absolute; clip: rect(2.974em 1001.54em 4.664em -999.997em); top: -4.762em; left: 0em;"><span class="mtd" id="MathJax-Span-2466"><span class="mrow" id="MathJax-Span-2467"><span class="mfrac" id="MathJax-Span-2468"><span style="display: inline-block; position: relative; width: 1.335em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-2469" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.18em 4.255em -999.997em); top: -3.584em; left: 50%; margin-left: -0.612em;"><span class="mrow" id="MathJax-Span-2470"><span class="mi" id="MathJax-Span-2471" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-2472" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-2473" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.34em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.335em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.02em; left: 0em;"><span class="mtd" id="MathJax-Span-2482"><span class="mrow" id="MathJax-Span-2483"><span class="mn" id="MathJax-Span-2484" style="font-family: STIXGeneral;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.359em 1005.18em 5.126em -999.997em); top: -3.993em; left: 2.666em;"><span style="display: inline-block; position: relative; width: 5.177em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1005.18em 4.152em -999.997em); top: -4.762em; left: 0em;"><span class="mtd" id="MathJax-Span-2474"><span class="mrow" id="MathJax-Span-2475"><span class="mspace" id="MathJax-Span-2476" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-2477" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-2478" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mi" id="MathJax-Span-2479" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2480" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mi" id="MathJax-Span-2481" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">b</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1005.02em 4.152em -999.997em); top: -3.02em; left: 0em;"><span class="mtd" id="MathJax-Span-2485"><span class="mrow" id="MathJax-Span-2486"><span class="mspace" id="MathJax-Span-2487" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-2488" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-2489" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-2490" style="font-family: STIXGeneral; font-style: italic;">h</span><span class="mi" id="MathJax-Span-2491" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mi" id="MathJax-Span-2492" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-2493" style="font-family: STIXGeneral; font-style: italic;">w</span><span class="mi" id="MathJax-Span-2494" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-2495" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-2496" style="font-family: STIXGeneral; font-style: italic;">e</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2497"></span></span></span><span style="display: inline-block; width: 0px; height: 4.152em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.622em; border-left: 0px solid; width: 0px; height: 3.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mfrac><mn>1</mn><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mtd><mtd><mspace width="1em"></mspace><mi>a</mi><mo><</mo><mi>x</mi><mo><</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mspace width="1em"></mspace><mi>o</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>w</mi><mi>i</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-172">f(x) = \left\{
\begin{array}{ll}
\frac{1}{b-a} & \quad a < x < b \\
0 & \quad otherwise
\end{array}
\right.</script></span><br>
and</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-173-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>0</mn></mtd><mtd><mspace width="1em" /><mi>x</mi><mo>&#x2264;</mo><mi>a</mi></mtd></mtr><mtr><mtd><mfrac><mrow><mi>x</mi><mo>&#x2212;</mo><mi>a</mi></mrow><mrow><mi>b</mi><mo>&#x2212;</mo><mi>a</mi></mrow></mfrac></mtd><mtd><mspace width="1em" /><mi>a</mi><mo>&#x2264;</mo><mi>x</mi><mo>&#x2264;</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mspace width="1em" /><mi>x</mi><mo>&#x2265;</mo><mi>b</mi></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2498" style="width: 14.449em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.837em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.257em 1011.84em 6.867em -999.997em); top: -4.813em; left: 0em;"><span class="mrow" id="MathJax-Span-2499"><span class="mi" id="MathJax-Span-2500" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2501" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2502" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2503" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2504" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mrow" id="MathJax-Span-2505" style="padding-left: 0.31em;"><span class="mo" id="MathJax-Span-2506" style="vertical-align: 2.41em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; font-family: STIXSizeOneSym; top: -3.276em; left: 0em;">⎧<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXSizeOneSym; top: 0.003em; left: 0em;">⎩<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXSizeOneSym; top: -1.637em; left: 0em;">⎨<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXSizeOneSym; position: absolute; top: -2.456em; left: 0em;">⎪<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXSizeOneSym; position: absolute; top: -0.817em; left: 0em;">⎪<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mtable" id="MathJax-Span-2507" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 7.841em; height: 0px;"><span style="position: absolute; clip: rect(2.564em 1001.54em 6.816em -999.997em); top: -4.967em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -5.581em; left: 0em;"><span class="mtd" id="MathJax-Span-2508"><span class="mrow" id="MathJax-Span-2509"><span class="mn" id="MathJax-Span-2510" style="font-family: STIXGeneral;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.974em 1001.54em 4.664em -999.997em); top: -4.045em; left: 0em;"><span class="mtd" id="MathJax-Span-2517"><span class="mrow" id="MathJax-Span-2518"><span class="mfrac" id="MathJax-Span-2519"><span style="display: inline-block; position: relative; width: 1.335em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.435em 1001.13em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.561em;"><span class="mrow" id="MathJax-Span-2520"><span class="mi" id="MathJax-Span-2521" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2522" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-2523" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.18em 4.255em -999.997em); top: -3.584em; left: 50%; margin-left: -0.612em;"><span class="mrow" id="MathJax-Span-2524"><span class="mi" id="MathJax-Span-2525" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-2526" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-2527" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.34em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.335em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -2.303em; left: 0em;"><span class="mtd" id="MathJax-Span-2536"><span class="mrow" id="MathJax-Span-2537"><span class="mn" id="MathJax-Span-2538" style="font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.972em;"></span></span><span style="position: absolute; clip: rect(2.615em 1005.18em 6.919em -999.997em); top: -4.967em; left: 2.666em;"><span style="display: inline-block; position: relative; width: 5.177em; height: 0px;"><span style="position: absolute; clip: rect(3.23em 1003.33em 4.255em -999.997em); top: -5.581em; left: 0em;"><span class="mtd" id="MathJax-Span-2511"><span class="mrow" id="MathJax-Span-2512"><span class="mspace" id="MathJax-Span-2513" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-2514" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2515" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mi" id="MathJax-Span-2516" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">a</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1005.18em 4.255em -999.997em); top: -4.045em; left: 0em;"><span class="mtd" id="MathJax-Span-2528"><span class="mrow" id="MathJax-Span-2529"><span class="mspace" id="MathJax-Span-2530" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-2531" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-2532" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mi" id="MathJax-Span-2533" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2534" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mi" id="MathJax-Span-2535" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">b</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1003.33em 4.255em -999.997em); top: -2.303em; left: 0em;"><span class="mtd" id="MathJax-Span-2539"><span class="mrow" id="MathJax-Span-2540"><span class="mspace" id="MathJax-Span-2541" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-2542" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2543" style="font-family: STIXGeneral; padding-left: 0.31em;">≥</span><span class="mi" id="MathJax-Span-2544" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">b</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.972em;"></span></span></span></span><span class="mo" id="MathJax-Span-2545"></span></span></span><span style="display: inline-block; width: 0px; height: 4.818em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.372em; border-left: 0px solid; width: 0px; height: 5.378em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>0</mn></mtd><mtd><mspace width="1em"></mspace><mi>x</mi><mo>≤</mo><mi>a</mi></mtd></mtr><mtr><mtd><mfrac><mrow><mi>x</mi><mo>−</mo><mi>a</mi></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mtd><mtd><mspace width="1em"></mspace><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mspace width="1em"></mspace><mi>x</mi><mo>≥</mo><mi>b</mi></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-173">F(x) = \left\{
\begin{array}{ll}
0 & \quad x \leq a \\
\frac{x-a}{b-a} & \quad a \leq x \leq b \\
1 & \quad x \geq b
\end{array}
\right.</script></span></p><h4 id="mean-and-variance-of-a-rectangular-distribution"><a class="anchor hidden-xs" href="#mean-and-variance-of-a-rectangular-distribution" title="mean-and-variance-of-a-rectangular-distribution"><span class="octicon octicon-link"></span></a>Mean and variance of a rectangular distribution</h4><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-174-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>&#x03BC;</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2546" style="width: 10.71em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.763em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1008.71em 3.384em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2547"><span class="mi" id="MathJax-Span-2548" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2549" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2550" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2551" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2552" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2553" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">μ</span><span class="mo" id="MathJax-Span-2554" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2555" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-2556" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-2557" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-2558" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2559" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-2560" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-2561" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mo" id="MathJax-Span-2562" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.872em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>μ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-174">E(X) = \mu = \frac{1}{2}(a+b)</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-175-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>&#x03C3;</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>12</mn></mfrac><mo stretchy="false">(</mo><mi>b</mi><mo>&#x2212;</mo><mi>a</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2563" style="width: 13.681em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.222em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1011.22em 3.384em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2564"><span class="mi" id="MathJax-Span-2565" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-2566" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-2567" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2568" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2569" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2570" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2571" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-2572" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2573" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.617em;"><span class="mn" id="MathJax-Span-2574" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2575" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2576" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-2577" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mn" id="MathJax-Span-2578" style="font-family: STIXGeneral;">12</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.13em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.13em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-2579" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2580" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-2581" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-2582" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">a</span><span class="msubsup" id="MathJax-Span-2583"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2584" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-2585" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.872em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>12</mn></mfrac><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-175">Var(X) = \sigma^2 = \frac{1}{12}(b-a)^2 </script></span></p><p>Given the mean and the variance of a rectangular distribution, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-176-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2586" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2587"><span class="mi" id="MathJax-Span-2588" style="font-family: STIXGeneral; font-style: italic;">a</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-176">a</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-177-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2589" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2590"><span class="mi" id="MathJax-Span-2591" style="font-family: STIXGeneral; font-style: italic;">b</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></span></span><script type="math/tex" id="MathJax-Element-177">b</script></span> can be found by solving simultaneously.</p><h2 id="estimation"><a class="anchor hidden-xs" href="#estimation" title="estimation"><span class="octicon octicon-link"></span></a>Estimation</h2><h3 id="mean"><a class="anchor hidden-xs" href="#mean" title="mean"><span class="octicon octicon-link"></span></a>Mean</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-178-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mo>&#x2211;</mo><mi>x</mi></mrow><mi>n</mi></mfrac></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mfrac><mrow><mo>&#x2211;</mo><mi>f</mi><mi>x</mi></mrow><mrow><mo>&#x2211;</mo><mi>f</mi></mrow></mfrac></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2592" style="width: 5.433em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.46em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-0.715em 1004.31em 4.613em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2593"><span class="mtable" id="MathJax-Span-2594" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 4.152em; height: 0px;"><span style="position: absolute; clip: rect(2.154em 1000.51em 5.791em -999.997em); top: -4.352em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.025em 1000.51em 4.152em -999.997em); top: -5.274em; right: 0em;"><span class="mtd" id="MathJax-Span-2595"><span class="mrow" id="MathJax-Span-2596"><span class="munderover" id="MathJax-Span-2597"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-2598" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.46em 3.998em -999.997em); top: -4.506em; left: 0.054em;"><span class="mo" id="MathJax-Span-2599" style=""><span style="display: inline-block; position: relative; width: 0.464em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.259em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -2.713em; right: 0em;"><span class="mtd" id="MathJax-Span-2609"><span class="mrow" id="MathJax-Span-2610"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.357em;"></span></span><span style="position: absolute; clip: rect(3.076em 1003.64em 8.404em -999.997em); top: -5.991em; left: 0.515em;"><span style="display: inline-block; position: relative; width: 3.64em; height: 0px;"><span style="position: absolute; clip: rect(2.359em 1003.23em 4.869em -999.997em); top: -5.274em; left: 0em;"><span class="mtd" id="MathJax-Span-2600"><span class="mrow" id="MathJax-Span-2601"><span class="mi" id="MathJax-Span-2602"></span><span class="mo" id="MathJax-Span-2603" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2604" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.693em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.076em 1001.59em 4.408em -999.997em); top: -4.711em; left: 50%; margin-left: -0.766em;"><span class="mrow" id="MathJax-Span-2605"><span class="mo" id="MathJax-Span-2606" style="font-family: STIXGeneral; vertical-align: 0.003em;">∑</span><span class="mi" id="MathJax-Span-2607" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-2608" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.69em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.693em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.359em 1003.64em 5.126em -999.997em); top: -2.713em; left: 0em;"><span class="mtd" id="MathJax-Span-2611"><span class="mrow" id="MathJax-Span-2612"><span class="mi" id="MathJax-Span-2613"></span><span class="mo" id="MathJax-Span-2614" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2615" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.103em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.076em 1002em 4.408em -999.997em); top: -4.711em; left: 50%; margin-left: -0.971em;"><span class="mrow" id="MathJax-Span-2616"><span class="mo" id="MathJax-Span-2617" style="font-family: STIXGeneral; vertical-align: 0.003em;">∑</span><span class="mi" id="MathJax-Span-2618" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mi" id="MathJax-Span-2619" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.076em 1001.54em 4.408em -999.997em); top: -3.276em; left: 50%; margin-left: -0.766em;"><span class="mrow" id="MathJax-Span-2620"><span class="mo" id="MathJax-Span-2621" style="font-family: STIXGeneral; vertical-align: 0.003em;">∑</span><span class="mi" id="MathJax-Span-2622" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.1em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.103em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.996em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.809em; border-left: 0px solid; width: 0px; height: 6.253em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mover><mi>x</mi><mo accent="false">¯</mo></mover></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mo>∑</mo><mi>x</mi></mrow><mi>n</mi></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mo>∑</mo><mi>f</mi><mi>x</mi></mrow><mrow><mo>∑</mo><mi>f</mi></mrow></mfrac></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-178">\begin{align} \overline{x} &= \frac{\sum x}{n} \\ &= \frac{\sum fx}{\sum f} \end{align}</script></span></p><h3 id="sample-variance"><a class="anchor hidden-xs" href="#sample-variance" title="sample-variance"><span class="octicon octicon-link"></span></a>Sample variance</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-179-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mo stretchy="false">(</mo><msub><mi>&#x03C3;</mi><mi>n</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>&#x2212;</mo><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mi>n</mi></mfrac><mo>&#x2212;</mo><msup><mi>&#x03BC;</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mfrac><mrow><mo>&#x2211;</mo><mi>f</mi><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mo>&#x2211;</mo><mi>f</mi></mrow></mfrac><mo>&#x2212;</mo><msup><mi>&#x03BC;</mi><mn>2</mn></msup></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2623" style="width: 10.505em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.609em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-2.047em 1008.46em 5.945em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2624"><span class="mtable" id="MathJax-Span-2625" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 8.302em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1002em 9.787em -999.997em); top: -7.016em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.001em; height: 0px;"><span style="position: absolute; clip: rect(2.974em 1002em 4.306em -999.997em); top: -6.555em; right: 0em;"><span class="mtd" id="MathJax-Span-2626"><span class="mrow" id="MathJax-Span-2627"><span class="mo" id="MathJax-Span-2628" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-2629"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2630" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2631" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2632"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2633" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-2634" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -3.993em; right: 0em;"><span class="mtd" id="MathJax-Span-2651"><span class="mrow" id="MathJax-Span-2652"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -1.381em; right: 0em;"><span class="mtd" id="MathJax-Span-2666"><span class="mrow" id="MathJax-Span-2667"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 7.021em;"></span></span><span style="position: absolute; clip: rect(4.408em 1006.3em 12.4em -999.997em); top: -8.655em; left: 2.001em;"><span style="display: inline-block; position: relative; width: 6.304em; height: 0px;"><span style="position: absolute; clip: rect(2.308em 1004.92em 4.869em -999.997em); top: -6.555em; left: 0em;"><span class="mtd" id="MathJax-Span-2635"><span class="mrow" id="MathJax-Span-2636"><span class="mi" id="MathJax-Span-2637"></span><span class="mo" id="MathJax-Span-2638" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2639" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 3.384em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1003.23em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.637em;"><span class="mrow" id="MathJax-Span-2640"><span class="mo" id="MathJax-Span-2641" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2642" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2643" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="munderover" id="MathJax-Span-2644" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-2645" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.46em 3.998em -999.997em); top: -4.506em; left: 0.054em;"><span class="mo" id="MathJax-Span-2646" style=""><span style="display: inline-block; position: relative; width: 0.464em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.259em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2647"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2648" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.361em;"><span class="mn" id="MathJax-Span-2649" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-2650" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1003.38em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.384em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.308em 1004.77em 4.869em -999.997em); top: -3.993em; left: 0em;"><span class="mtd" id="MathJax-Span-2653"><span class="mrow" id="MathJax-Span-2654"><span class="mi" id="MathJax-Span-2655"></span><span class="mo" id="MathJax-Span-2656" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2657" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1000.92em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="msubsup" id="MathJax-Span-2658"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2659" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="mn" id="MathJax-Span-2660" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-2661" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.03em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.027em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-2662" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-2663" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2664" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mn" id="MathJax-Span-2665" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.257em 1006.3em 5.126em -999.997em); top: -1.381em; left: 0em;"><span class="mtd" id="MathJax-Span-2668"><span class="mrow" id="MathJax-Span-2669"><span class="mi" id="MathJax-Span-2670"></span><span class="mo" id="MathJax-Span-2671" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2672" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.615em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1002.46em 4.408em -999.997em); top: -4.711em; left: 50%; margin-left: -1.227em;"><span class="mrow" id="MathJax-Span-2673"><span class="mo" id="MathJax-Span-2674" style="font-family: STIXGeneral; vertical-align: 0.003em;">∑</span><span class="mi" id="MathJax-Span-2675" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="msubsup" id="MathJax-Span-2676"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2677" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="mn" id="MathJax-Span-2678" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.076em 1001.54em 4.408em -999.997em); top: -3.276em; left: 50%; margin-left: -0.766em;"><span class="mrow" id="MathJax-Span-2679"><span class="mo" id="MathJax-Span-2680" style="font-family: STIXGeneral; vertical-align: 0.003em;">∑</span><span class="mi" id="MathJax-Span-2681" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.615em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-2682" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-2683" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2684" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mn" id="MathJax-Span-2685" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 8.66em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -4.434em; border-left: 0px solid; width: 0px; height: 9.503em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mo stretchy="false">(</mo><msub><mi>σ</mi><mi>n</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mover><mi>x</mi><mo accent="false">¯</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mi>n</mi></mfrac><mo>−</mo><msup><mi>μ</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mo>∑</mo><mi>f</mi><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mo>∑</mo><mi>f</mi></mrow></mfrac><mo>−</mo><msup><mi>μ</mi><mn>2</mn></msup></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-179">\begin{align} (\sigma _n)^2 &= \frac{(x-\overline{x})^2}{n} \\
&= \frac{x^2}{n} - \mu ^2 \\
&= \frac{\sum f x^2}{\sum f} - \mu ^2 \end{align}</script></span></p><h3 id="unbiased-estimator-of-the-population-variance"><a class="anchor hidden-xs" href="#unbiased-estimator-of-the-population-variance" title="unbiased-estimator-of-the-population-variance"><span class="octicon octicon-link"></span></a>Unbiased estimator of the population variance</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-180-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><msub><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mfrac><mi>n</mi><mrow><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></mfrac><mo>&#x00D7;</mo><mo stretchy="false">(</mo><msub><mi>&#x03C3;</mi><mi>n</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2686" style="width: 12.041em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.839em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1009.84em 3.537em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2687"><span class="mo" id="MathJax-Span-2688" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-2689"><span style="display: inline-block; position: relative; width: 1.744em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2690" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="texatom" id="MathJax-Span-2691"><span class="mrow" id="MathJax-Span-2692"><span class="mi" id="MathJax-Span-2693" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2694" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2695" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2696"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2697" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-2698" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2699" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2700" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-2701" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1002.1em 4.306em -999.997em); top: -3.327em; left: 50%; margin-left: -1.124em;"><span class="mrow" id="MathJax-Span-2702"><span class="mi" id="MathJax-Span-2703" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2704" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-2705" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.31em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.308em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-2706" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mo" id="MathJax-Span-2707" style="font-family: STIXGeneral; padding-left: 0.259em;">(</span><span class="msubsup" id="MathJax-Span-2708"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2709" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2710" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2711"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2712" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-2713" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.059em; border-left: 0px solid; width: 0px; height: 2.503em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><msub><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mfrac><mi>n</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>×</mo><mo stretchy="false">(</mo><msub><mi>σ</mi><mi>n</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-180">(\sigma _{n-1})^2 = \frac{n}{n-1} \times (\sigma _n)^2</script></span></p><h3 id="confidence-intervals"><a class="anchor hidden-xs" href="#confidence-intervals" title="confidence-intervals"><span class="octicon octicon-link"></span></a>Confidence intervals</h3><h4 id="interpretation-of-confidence-intervals"><a class="anchor hidden-xs" href="#interpretation-of-confidence-intervals" title="interpretation-of-confidence-intervals"><span class="octicon octicon-link"></span></a>Interpretation of confidence intervals</h4><p>For a <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-181-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2714" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2715"><span class="mi" id="MathJax-Span-2716" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-181">k</script></span> percent confidence interval for the mean, across a number of different samples <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-182-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2717" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2718"><span class="mi" id="MathJax-Span-2719" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-182">k</script></span> percent of the confidence intervals for the mean will contain the true population mean.</p><h4 id="writing-confidence-intervals"><a class="anchor hidden-xs" href="#writing-confidence-intervals" title="writing-confidence-intervals"><span class="octicon octicon-link"></span></a>Writing confidence intervals</h4><p>Confidence intervals should be written to a high degree of accuracy in the format (lower limit, upper limit)</p><h4 id="calculating-confidence-intervals"><a class="anchor hidden-xs" href="#calculating-confidence-intervals" title="calculating-confidence-intervals"><span class="octicon octicon-link"></span></a>Calculating confidence intervals</h4><p>If the population variance is given for a normally distributed population the Z tables are used and the confidence interval is given by:</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-183-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><mtext>&#xA0;</mtext><mo>&#x00B1;</mo><mtext>&#xA0;</mtext><msub><mi>Z</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B1;</mi></mrow></msub><mo>&#x00D7;</mo><msqrt><mfrac><msup><mi>&#x03C3;</mi><mn>2</mn></msup><mi>n</mi></mfrac></msqrt></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2720" style="width: 8.455em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.919em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1006.92em 3.589em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2721"><span class="munderover" id="MathJax-Span-2722"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-2723" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.46em 3.998em -999.997em); top: -4.506em; left: 0.054em;"><span class="mo" id="MathJax-Span-2724" style=""><span style="display: inline-block; position: relative; width: 0.464em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.259em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mtext" id="MathJax-Span-2725" style="font-family: STIXGeneral;"> </span><span class="mo" id="MathJax-Span-2726" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="mtext" id="MathJax-Span-2727" style="font-family: STIXGeneral; padding-left: 0.259em;"> </span><span class="msubsup" id="MathJax-Span-2728"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2729" style="font-family: STIXGeneral; font-style: italic;">Z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.566em;"><span class="texatom" id="MathJax-Span-2730"><span class="mrow" id="MathJax-Span-2731"><span class="mi" id="MathJax-Span-2732" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">α</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2733" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msqrt" id="MathJax-Span-2734" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.513em; height: 0px;"><span style="position: absolute; clip: rect(2.41em 1001.39em 4.869em -999.997em); top: -3.993em; left: 1.13em;"><span class="mrow" id="MathJax-Span-2735"><span class="mfrac" id="MathJax-Span-2736"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.076em 1001.03em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.51em;"><span class="msubsup" id="MathJax-Span-2737"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2738" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.617em;"><span class="mn" id="MathJax-Span-2739" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-2740" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.13em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.13em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1001.39em 3.384em -999.997em); top: -4.711em; left: 1.13em;"><span style="display: inline-block; position: relative; width: 1.386em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0.925em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.464em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(1.796em 1001.18em 4.562em -999.997em); top: -3.532em; left: 0em;"><span style="font-family: STIXSizeTwoSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.122em; border-left: 0px solid; width: 0px; height: 3.128em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mover><mi>x</mi><mo accent="false">¯</mo></mover><mtext> </mtext><mo>±</mo><mtext> </mtext><msub><mi>Z</mi><mrow class="MJX-TeXAtom-ORD"><mi>α</mi></mrow></msub><mo>×</mo><msqrt><mfrac><msup><mi>σ</mi><mn>2</mn></msup><mi>n</mi></mfrac></msqrt></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-183">\overline{x} \ \pm \ Z_{\alpha} \times \sqrt{\frac{\sigma ^2}{n}}</script></span></p><p>If the population variance is unknown, but the sample size is greater than 30 the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-184-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2741" style="width: 0.771em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.62em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2742"><span class="mi" id="MathJax-Span-2743" style="font-family: STIXGeneral; font-style: italic;">Z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script type="math/tex" id="MathJax-Element-184">Z</script></span> tables are used due to central limit theorem. The above formula is still used, with the following assumptions:</p><ul>
<li>The data can be regarded as a random sample</li>
<li>Central limit theorem allows the distribution to be assumed to follow a normal distribution</li>
</ul><p>If the population variance is unknown, and the sample size is less than <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-185-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2744" style="width: 1.283em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.03em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2745"><span class="mn" id="MathJax-Span-2746" style="font-family: STIXGeneral;">30</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math></span></span><script type="math/tex" id="MathJax-Element-185">30</script></span>, the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-186-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2747" style="width: 0.31em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.259em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.847em 1000.26em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2748"><span class="mi" id="MathJax-Span-2749" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.816em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span><script type="math/tex" id="MathJax-Element-186">t</script></span> tables are used with <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-187-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BD;</mi><mo>=</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2750" style="width: 4.818em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.947em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.85em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2751"><span class="mi" id="MathJax-Span-2752" style="font-family: STIXGeneral; font-style: italic;">ν</span><span class="mo" id="MathJax-Span-2753" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2754" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">n</span><span class="mo" id="MathJax-Span-2755" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-2756" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ν</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-187">\nu = n - 1</script></span> degrees of freedom. The confidence interval is then given by:</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-188-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><mtext>&#xA0;</mtext><mo>&#x00B1;</mo><mtext>&#xA0;</mtext><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B1;</mi></mrow></msub><mo>&#x00D7;</mo><msqrt><mfrac><msup><mi>S</mi><mn>2</mn></msup><mi>n</mi></mfrac></msqrt></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2757" style="width: 8.148em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.662em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1006.66em 3.589em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2758"><span class="munderover" id="MathJax-Span-2759"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-2760" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.46em 3.998em -999.997em); top: -4.506em; left: 0.054em;"><span class="mo" id="MathJax-Span-2761" style=""><span style="display: inline-block; position: relative; width: 0.464em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.259em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mtext" id="MathJax-Span-2762" style="font-family: STIXGeneral;"> </span><span class="mo" id="MathJax-Span-2763" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="mtext" id="MathJax-Span-2764" style="font-family: STIXGeneral; padding-left: 0.259em;"> </span><span class="msubsup" id="MathJax-Span-2765"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2766" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.259em;"><span class="texatom" id="MathJax-Span-2767"><span class="mrow" id="MathJax-Span-2768"><span class="mi" id="MathJax-Span-2769" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">α</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2770" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msqrt" id="MathJax-Span-2771" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.513em; height: 0px;"><span style="position: absolute; clip: rect(2.41em 1001.34em 4.869em -999.997em); top: -3.993em; left: 1.13em;"><span class="mrow" id="MathJax-Span-2772"><span class="mfrac" id="MathJax-Span-2773"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.076em 1000.98em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.51em;"><span class="msubsup" id="MathJax-Span-2774"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2775" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.566em;"><span class="mn" id="MathJax-Span-2776" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-2777" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.13em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.13em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1001.39em 3.384em -999.997em); top: -4.711em; left: 1.13em;"><span style="display: inline-block; position: relative; width: 1.386em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0.873em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.412em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(1.796em 1001.18em 4.562em -999.997em); top: -3.532em; left: 0em;"><span style="font-family: STIXSizeTwoSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.122em; border-left: 0px solid; width: 0px; height: 3.128em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mover><mi>x</mi><mo accent="false">¯</mo></mover><mtext> </mtext><mo>±</mo><mtext> </mtext><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mi>α</mi></mrow></msub><mo>×</mo><msqrt><mfrac><msup><mi>S</mi><mn>2</mn></msup><mi>n</mi></mfrac></msqrt></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-188">\overline{x} \ \pm \ t_{\alpha} \times \sqrt{\frac{S^2}{n}}</script></span></p><p>Example</p><blockquote>
<p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-278-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3945" style="width: 1.22em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.986em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1000.94em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3946"><span class="mn" id="MathJax-Span-3947" style="font-family: STIXGeneral;">20</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math></span></span><script type="math/tex" id="MathJax-Element-278">20</script></span> bottles are selected from a production line. The volume of liquid in each is recorded as <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-279-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3948" style="width: 0.611em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.471em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.97em 1000.47em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3949"><span class="mi" id="MathJax-Span-3950" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.689em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-279">x</script></span> ml<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-280-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>&#x2211;</mo><mi>x</mi><mo>=</mo><mn>1518.9</mn><mspace width="1em" /><mo>&#x2211;</mo><mo stretchy="false">(</mo><mi>x</mi><mo>&#x2212;</mo><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mn>7.2895</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3951" style="width: 19.44em; display: inline-block;"><span style="display: inline-block; position: relative; width: 15.927em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.829em 1015.88em 3.562em -999.998em); top: -2.948em; left: 0em;"><span class="mrow" id="MathJax-Span-3952"><span class="mo" id="MathJax-Span-3953" style="font-family: STIXSizeOneSym; vertical-align: -0.513em;">∑</span><span class="mi" id="MathJax-Span-3954" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.19em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3955" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-3956" style="font-family: STIXGeneral; padding-left: 0.33em;">1518.9</span><span class="mspace" id="MathJax-Span-3957" style="height: 0em; vertical-align: 0em; width: 1.126em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-3958" style="font-family: STIXSizeOneSym; vertical-align: -0.513em; padding-left: 0.19em;">∑</span><span class="mo" id="MathJax-Span-3959" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-3960" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-3961" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="munderover" id="MathJax-Span-3962" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px;"><span style="position: absolute; clip: rect(3.422em 1000.47em 4.124em -999.998em); top: -3.979em; left: 0.002em;"><span class="mi" id="MathJax-Span-3963" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.562em 1000.47em 3.984em -999.998em); top: -4.494em; left: 0.049em;"><span class="mo" id="MathJax-Span-3964" style=""><span style="display: inline-block; position: relative; width: 0.471em; height: 0px;"><span style="position: absolute; top: -3.979em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.979em; left: 0.237em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.979em; left: 0.096em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-3965"><span style="display: inline-block; position: relative; width: 0.752em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.28em 4.311em -999.998em); top: -3.979em; left: 0em;"><span class="mo" id="MathJax-Span-3966" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.4em; left: 0.33em;"><span class="mn" id="MathJax-Span-3967" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-3968" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-3969" style="font-family: STIXGeneral; padding-left: 0.33em;">7.2895</span></span><span style="display: inline-block; width: 0px; height: 2.953em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.626em; border-left: 0px solid; width: 0px; height: 1.889em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>∑</mo><mi>x</mi><mo>=</mo><mn>1518.9</mn><mspace width="1em"></mspace><mo>∑</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mover><mi>x</mi><mo accent="false">¯</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mn>7.2895</mn></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-280">\sum x = 1518.9 \quad \sum (x - \overline{x})^2 = 7.2895</script></span><br>
Stating any assumptions made, construct a <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-281-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>95</mn><mi mathvariant="normal">&#x0025;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3970" style="width: 2.11em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.735em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1001.69em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-3971"><span class="mn" id="MathJax-Span-3972" style="font-family: STIXGeneral;">95</span><span class="mi" id="MathJax-Span-3973" style="font-family: STIXGeneral;">%</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 1.031em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>95</mn><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-281">95\%</script></span> confidence interval for the mean.<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-282-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>n</mi><mo>=</mo><mn>20</mn><mspace linebreak="newline" /><mi>m</mi><mi>e</mi><mi>a</mi><mi>n</mi><mo>=</mo><mfrac><mrow><mo>&#x2211;</mo><mi>x</mi></mrow><mi>n</mi></mfrac><mspace linebreak="newline" /><mi>s</mi><mi>a</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>e</mi><mtext>&#xA0;</mtext><mi>v</mi><mi>a</mi><mi>r</mi><mi>i</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi><mo>=</mo><mfrac><mrow><mo>&#x2211;</mo><mo stretchy="false">(</mo><mi>x</mi><mo>&#x2212;</mo><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac><mspace linebreak="newline" /><mi>u</mi><mi>n</mi><mi>b</mi><mi>i</mi><mi>a</mi><mi>s</mi><mi>e</mi><mi>d</mi><mtext>&#xA0;</mtext><mi>e</mi><mi>s</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi><mtext>&#xA0;</mtext><mi>o</mi><mi>f</mi><mtext>&#xA0;</mtext><mi>p</mi><mi>o</mi><mi>p</mi><mi>u</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext>&#xA0;</mtext><mi>v</mi><mi>a</mi><mi>r</mi><mi>i</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi><mo>=</mo><mfrac><mrow><mo>&#x2211;</mo><mo stretchy="false">(</mo><mi>x</mi><mo>&#x2212;</mo><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3974" style="width: 100%; display: inline-block; min-width: 28.199em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(9.183em 1023.09em 18.41em -999.998em); top: -9.974em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-3975"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1002.81em 4.124em -999.998em); top: -3.979em; left: 50%; margin-left: -1.403em;"><span class="mi" id="MathJax-Span-3976" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-3977" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-3978" style="font-family: STIXGeneral; padding-left: 0.33em;">20</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.344em 1005.44em 4.827em -999.998em); top: -2.012em; left: 50%; margin-left: -2.714em;"><span class="mspace" id="MathJax-Span-3979" style="height: 0em; vertical-align: 0em; width: 0em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-3980" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="mi" id="MathJax-Span-3981" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mi" id="MathJax-Span-3982" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-3983" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-3984" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mfrac" id="MathJax-Span-3985" style="padding-left: 0.33em;"><span style="display: inline-block; position: relative; width: 1.689em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3.094em 1001.55em 4.405em -999.998em); top: -4.681em; left: 50%; margin-left: -0.794em;"><span class="mrow" id="MathJax-Span-3986"><span class="mo" id="MathJax-Span-3987" style="font-family: STIXGeneral; vertical-align: 0.002em;">∑</span><span class="mi" id="MathJax-Span-3988" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.19em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.422em 1000.47em 4.124em -999.998em); top: -3.276em; left: 50%; margin-left: -0.232em;"><span class="mi" id="MathJax-Span-3989" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1001.69em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.689em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.297em 1012.41em 4.827em -999.998em); top: 0.752em; left: 50%; margin-left: -6.227em;"><span class="mspace" id="MathJax-Span-3990" style="height: 0em; vertical-align: 0em; width: 0em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-3991" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-3992" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-3993" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="mi" id="MathJax-Span-3994" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mi" id="MathJax-Span-3995" style="font-family: STIXGeneral; font-style: italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-3996" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mtext" id="MathJax-Span-3997" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-3998" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mi" id="MathJax-Span-3999" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-4000" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-4001" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-4002" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-4003" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-4004" style="font-family: STIXGeneral; font-style: italic;">c</span><span class="mi" id="MathJax-Span-4005" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mo" id="MathJax-Span-4006" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mfrac" id="MathJax-Span-4007" style="padding-left: 0.33em;"><span style="display: inline-block; position: relative; width: 4.218em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3em 1004.12em 4.405em -999.998em); top: -4.681em; left: 50%; margin-left: -2.059em;"><span class="mrow" id="MathJax-Span-4008"><span class="mo" id="MathJax-Span-4009" style="font-family: STIXGeneral; vertical-align: 0.002em;">∑</span><span class="mo" id="MathJax-Span-4010" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-4011" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-4012" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="munderover" id="MathJax-Span-4013" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px;"><span style="position: absolute; clip: rect(3.422em 1000.47em 4.124em -999.998em); top: -3.979em; left: 0.002em;"><span class="mi" id="MathJax-Span-4014" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.562em 1000.47em 3.984em -999.998em); top: -4.494em; left: 0.049em;"><span class="mo" id="MathJax-Span-4015" style=""><span style="display: inline-block; position: relative; width: 0.471em; height: 0px;"><span style="position: absolute; top: -3.979em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.979em; left: 0.237em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.979em; left: 0.096em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-4016"><span style="display: inline-block; position: relative; width: 0.752em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.28em 4.311em -999.998em); top: -3.979em; left: 0em;"><span class="mo" id="MathJax-Span-4017" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.354em; left: 0.33em;"><span class="mn" id="MathJax-Span-4018" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.422em 1000.47em 4.124em -999.998em); top: -3.276em; left: 50%; margin-left: -0.232em;"><span class="mi" id="MathJax-Span-4019" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1004.22em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.218em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(2.297em 1023.09em 4.967em -999.998em); top: 3.468em; left: 50%; margin-left: -11.567em;"><span class="mspace" id="MathJax-Span-4020" style="height: 0em; vertical-align: 0em; width: 0em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-4021" style="font-family: STIXGeneral; font-style: italic;">u</span><span class="mi" id="MathJax-Span-4022" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-4023" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mi" id="MathJax-Span-4024" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-4025" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-4026" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-4027" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mi" id="MathJax-Span-4028" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span class="mtext" id="MathJax-Span-4029" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-4030" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mi" id="MathJax-Span-4031" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-4032" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-4033" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-4034" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="mi" id="MathJax-Span-4035" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-4036" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-4037" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-4038" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mtext" id="MathJax-Span-4039" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-4040" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-4041" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.143em;"></span></span><span class="mtext" id="MathJax-Span-4042" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-4043" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mi" id="MathJax-Span-4044" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-4045" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mi" id="MathJax-Span-4046" style="font-family: STIXGeneral; font-style: italic;">u</span><span class="mi" id="MathJax-Span-4047" style="font-family: STIXGeneral; font-style: italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-4048" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-4049" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-4050" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-4051" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-4052" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mtext" id="MathJax-Span-4053" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-4054" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mi" id="MathJax-Span-4055" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-4056" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-4057" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-4058" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-4059" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-4060" style="font-family: STIXGeneral; font-style: italic;">c</span><span class="mi" id="MathJax-Span-4061" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mo" id="MathJax-Span-4062" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mfrac" id="MathJax-Span-4063" style="padding-left: 0.33em;"><span style="display: inline-block; position: relative; width: 4.218em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3em 1004.12em 4.405em -999.998em); top: -4.681em; left: 50%; margin-left: -2.059em;"><span class="mrow" id="MathJax-Span-4064"><span class="mo" id="MathJax-Span-4065" style="font-family: STIXGeneral; vertical-align: 0.002em;">∑</span><span class="mo" id="MathJax-Span-4066" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-4067" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-4068" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="munderover" id="MathJax-Span-4069" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 0.518em; height: 0px;"><span style="position: absolute; clip: rect(3.422em 1000.47em 4.124em -999.998em); top: -3.979em; left: 0.002em;"><span class="mi" id="MathJax-Span-4070" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.562em 1000.47em 3.984em -999.998em); top: -4.494em; left: 0.049em;"><span class="mo" id="MathJax-Span-4071" style=""><span style="display: inline-block; position: relative; width: 0.471em; height: 0px;"><span style="position: absolute; top: -3.979em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.979em; left: 0.237em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.979em; left: 0.096em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-4072"><span style="display: inline-block; position: relative; width: 0.752em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.28em 4.311em -999.998em); top: -3.979em; left: 0em;"><span class="mo" id="MathJax-Span-4073" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.354em; left: 0.33em;"><span class="mn" id="MathJax-Span-4074" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1002.06em 4.265em -999.998em); top: -3.276em; left: 50%; margin-left: -1.075em;"><span class="mrow" id="MathJax-Span-4075"><span class="mi" id="MathJax-Span-4076" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-4077" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="mn" id="MathJax-Span-4078" style="font-family: STIXGeneral; padding-left: 0.237em;">1</span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1004.22em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.218em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 9.979em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -10.169em; border-left: 0px solid; width: 0px; height: 11.031em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>n</mi><mo>=</mo><mn>20</mn><mspace linebreak="newline"></mspace><mi>m</mi><mi>e</mi><mi>a</mi><mi>n</mi><mo>=</mo><mfrac><mrow><mo>∑</mo><mi>x</mi></mrow><mi>n</mi></mfrac><mspace linebreak="newline"></mspace><mi>s</mi><mi>a</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>v</mi><mi>a</mi><mi>r</mi><mi>i</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi><mo>=</mo><mfrac><mrow><mo>∑</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mover><mi>x</mi><mo accent="false">¯</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac><mspace linebreak="newline"></mspace><mi>u</mi><mi>n</mi><mi>b</mi><mi>i</mi><mi>a</mi><mi>s</mi><mi>e</mi><mi>d</mi><mtext> </mtext><mi>e</mi><mi>s</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mi>p</mi><mi>o</mi><mi>p</mi><mi>u</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mi>v</mi><mi>a</mi><mi>r</mi><mi>i</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi><mo>=</mo><mfrac><mrow><mo>∑</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mover><mi>x</mi><mo accent="false">¯</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-282">n = 20 \\ mean = \frac{\sum x}{n} \\ sample\ variance = \frac{\sum (x - \overline{x})^2}{n} \\ unbiased\ estimator\ of\ population\ variance = \frac{\sum (x - \overline{x})^2}{n-1}</script></span><br>
Using <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-283-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BD;</mi><mo>=</mo><mn>20</mn><mo>&#x2212;</mo><mn>1</mn><mo>=</mo><mn>19</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4079" style="width: 8.293em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.794em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1006.75em 2.813em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-4080"><span class="mi" id="MathJax-Span-4081" style="font-family: STIXGeneral; font-style: italic;">ν</span><span class="mo" id="MathJax-Span-4082" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-4083" style="font-family: STIXGeneral; padding-left: 0.33em;">20</span><span class="mo" id="MathJax-Span-4084" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="mn" id="MathJax-Span-4085" style="font-family: STIXGeneral; padding-left: 0.237em;">1</span><span class="mo" id="MathJax-Span-4086" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-4087" style="font-family: STIXGeneral; padding-left: 0.33em;">19</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.226em; border-left: 0px solid; width: 0px; height: 1.089em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ν</mi><mo>=</mo><mn>20</mn><mo>−</mo><mn>1</mn><mo>=</mo><mn>19</mn></math></span></span><script type="math/tex" id="MathJax-Element-283">\nu = 20 - 1 = 19</script></span> degrees of freedom, the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-284-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>95</mn><mi mathvariant="normal">&#x0025;</mi><mtext>&#xA0;</mtext><mi>t</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4088" style="width: 2.813em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.297em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1002.3em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-4089"><span class="mn" id="MathJax-Span-4090" style="font-family: STIXGeneral;">95</span><span class="mi" id="MathJax-Span-4091" style="font-family: STIXGeneral;">%</span><span class="mtext" id="MathJax-Span-4092" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-4093" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 1.031em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>95</mn><mi mathvariant="normal">%</mi><mtext> </mtext><mi>t</mi></math></span></span><script type="math/tex" id="MathJax-Element-284">95\%\ t%</script></span> values is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-285-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2.093</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4094" style="width: 2.766em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.251em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.735em 1002.2em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-4095"><span class="mn" id="MathJax-Span-4096" style="font-family: STIXGeneral;">2.093</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2.093</mn></math></span></span><script type="math/tex" id="MathJax-Element-285">2.093</script></span><br>
The confidence interval is then given by<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-286-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>75.945</mn><mtext>&#xA0;</mtext><mo>&#x00B1;</mo><mtext>&#xA0;</mtext><mn>2.0893</mn><mo>&#x00D7;</mo><msqrt><mfrac><mn>0.38366</mn><mn>20</mn></mfrac></msqrt><mo>=</mo><mo stretchy="false">(</mo><mn>75.66</mn><mo>,</mo><mn>76.23</mn><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4097" style="width: 24.405em; display: inline-block;"><span style="display: inline-block; position: relative; width: 20.002em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.845em 1019.96em 3.609em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-4098"><span class="mn" id="MathJax-Span-4099" style="font-family: STIXGeneral;">75.945</span><span class="mtext" id="MathJax-Span-4100" style="font-family: STIXGeneral;"> </span><span class="mo" id="MathJax-Span-4101" style="font-family: STIXGeneral; padding-left: 0.237em;">±</span><span class="mtext" id="MathJax-Span-4102" style="font-family: STIXGeneral; padding-left: 0.237em;"> </span><span class="mn" id="MathJax-Span-4103" style="font-family: STIXGeneral;">2.0893</span><span class="mo" id="MathJax-Span-4104" style="font-family: STIXGeneral; padding-left: 0.237em;">×</span><span class="msqrt" id="MathJax-Span-4105" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 4.78em; height: 0px;"><span style="position: absolute; clip: rect(2.485em 1003.61em 4.827em -999.998em); top: -3.979em; left: 1.126em;"><span class="mrow" id="MathJax-Span-4106"><span class="mfrac" id="MathJax-Span-4107"><span style="display: inline-block; position: relative; width: 3.375em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3.141em 1003.23em 4.124em -999.998em); top: -4.635em; left: 50%; margin-left: -1.637em;"><span class="mn" id="MathJax-Span-4108" style="font-family: STIXGeneral;">0.38366</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1000.99em 4.124em -999.998em); top: -3.276em; left: 50%; margin-left: -0.513em;"><span class="mn" id="MathJax-Span-4109" style="font-family: STIXGeneral;">20</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1003.38em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.375em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3em 1003.66em 3.375em -999.998em); top: -4.681em; left: 1.126em;"><span style="display: inline-block; position: relative; width: 3.656em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.979em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.979em; left: 3.141em;">‾<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.979em; left: 0.424em;">‾<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.979em; left: 0.892em;">‾<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.979em; left: 1.314em;">‾<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.979em; left: 1.782em;">‾<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.979em; left: 2.251em;">‾<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.979em; left: 2.672em;">‾<span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(1.782em 1001.17em 4.546em -999.998em); top: -3.464em; left: 0em;"><span style="font-family: STIXSizeTwoSym;">√</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-4110" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mo" id="MathJax-Span-4111" style="font-family: STIXGeneral; padding-left: 0.33em;">(</span><span class="mn" id="MathJax-Span-4112" style="font-family: STIXGeneral;">75.66</span><span class="mo" id="MathJax-Span-4113" style="font-family: STIXGeneral;">,</span><span class="mn" id="MathJax-Span-4114" style="font-family: STIXGeneral; padding-left: 0.19em;">76.23</span><span class="mo" id="MathJax-Span-4115" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.197em; border-left: 0px solid; width: 0px; height: 3.146em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>75.945</mn><mtext> </mtext><mo>±</mo><mtext> </mtext><mn>2.0893</mn><mo>×</mo><msqrt><mfrac><mn>0.38366</mn><mn>20</mn></mfrac></msqrt><mo>=</mo><mo stretchy="false">(</mo><mn>75.66</mn><mo>,</mo><mn>76.23</mn><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-286">75.945\ \pm \ 2.0893 \times \sqrt{\frac{0.38366}{20}} = (75.66, 76.23)</script></span><br>
The assumptions made were that</p>
<ul>
<li>The contents of the bottles are normally distributed</li>
<li>The sample was selected randomly</li>
</ul>
</blockquote><h2 id="hypothesis-testing"><a class="anchor hidden-xs" href="#hypothesis-testing" title="hypothesis-testing"><span class="octicon octicon-link"></span></a>Hypothesis testing</h2><h3 id="procedure"><a class="anchor hidden-xs" href="#procedure" title="procedure"><span class="octicon octicon-link"></span></a>Procedure</h3><h4 id="step-1"><a class="anchor hidden-xs" href="#step-1" title="step-1"><span class="octicon octicon-link"></span></a>Step 1</h4><p>State the null hypothesis and the alternate hypothesis.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-198-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub><mo>:</mo><mtext>&#xA0;</mtext><mi>&#x03BC;</mi><mo>=</mo><mi>a</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2949" style="width: 5.638em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.613em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1004.61em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2950"><span class="msubsup" id="MathJax-Span-2951"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2952" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-2953" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2954" style="font-family: STIXGeneral; padding-left: 0.31em;">:</span><span class="mtext" id="MathJax-Span-2955" style="font-family: STIXGeneral; padding-left: 0.31em;"> </span><span class="mi" id="MathJax-Span-2956" style="font-family: STIXGeneral; font-style: italic;">μ</span><span class="mo" id="MathJax-Span-2957" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2958" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">a</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub><mo>:</mo><mtext> </mtext><mi>μ</mi><mo>=</mo><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-198">H_0:\ \mu = a</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-199-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub><mo>:</mo><mtext>&#xA0;</mtext><mi>&#x03BC;</mi><mo>&#x2260;</mo><mi>a</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2959" style="width: 5.638em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.613em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1004.61em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2960"><span class="msubsup" id="MathJax-Span-2961"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2962" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-2963" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2964" style="font-family: STIXGeneral; padding-left: 0.31em;">:</span><span class="mtext" id="MathJax-Span-2965" style="font-family: STIXGeneral; padding-left: 0.31em;"> </span><span class="mi" id="MathJax-Span-2966" style="font-family: STIXGeneral; font-style: italic;">μ</span><span class="mo" id="MathJax-Span-2967" style="font-family: STIXGeneral; padding-left: 0.31em;">≠</span><span class="mi" id="MathJax-Span-2968" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">a</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub><mo>:</mo><mtext> </mtext><mi>μ</mi><mo>≠</mo><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-199">H_1:\ \mu \neq a</script></span></p><p>Two tailed test</p><h4 id="step-2"><a class="anchor hidden-xs" href="#step-2" title="step-2"><span class="octicon octicon-link"></span></a>Step 2</h4><p>Choose the test statistic</p><p>For a known variance, or <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-200-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&gt;</mo><mn>30</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2969" style="width: 3.435em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.82em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.82em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2970"><span class="mi" id="MathJax-Span-2971" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2972" style="font-family: STIXGeneral; padding-left: 0.31em;">></span><span class="mn" id="MathJax-Span-2973" style="font-family: STIXGeneral; padding-left: 0.31em;">30</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>30</mn></math></span></span><script type="math/tex" id="MathJax-Element-200">n > 30</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-201-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>Z</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>&#x2212;</mo><mi>&#x03BC;</mi></mrow><msqrt><mfrac><msup><mi>&#x03C3;</mi><mn>2</mn></msup><mi>n</mi></mfrac></msqrt></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2974" style="width: 5.382em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.408em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1004.41em 4.511em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2975"><span class="mi" id="MathJax-Span-2976" style="font-family: STIXGeneral; font-style: italic;">Z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2977" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2978" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.281em 1002.15em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -1.073em;"><span class="mrow" id="MathJax-Span-2979"><span class="mi" id="MathJax-Span-2980" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2981" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-2982" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">μ</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.615em 1002.21em 4.767em -999.997em); top: -2.764em; left: 50%; margin-left: -1.073em;"><span class="msqrt" id="MathJax-Span-2983"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.08em 4.511em -999.997em); top: -3.993em; left: 1.078em;"><span class="mrow" id="MathJax-Span-2984"><span class="mfrac" id="MathJax-Span-2985"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.281em 1000.72em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.356em;"><span class="msubsup" id="MathJax-Span-2986"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2987" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.198em; left: 0.412em;"><span class="mn" id="MathJax-Span-2988" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-2989" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.87em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.873em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1001.13em 3.384em -999.997em); top: -4.403em; left: 1.078em;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0.617em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.31em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.308em 1001.08em 4.46em -999.997em); top: -3.686em; left: 0em;"><span style="font-family: STIXSizeOneSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.31em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.308em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.247em; border-left: 0px solid; width: 0px; height: 3.816em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>Z</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi>μ</mi></mrow><msqrt><mfrac><msup><mi>σ</mi><mn>2</mn></msup><mi>n</mi></mfrac></msqrt></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-201"> Z = \frac{x - \mu}{ \sqrt{\frac{\sigma ^2}{n}}}</script></span></p><p>For an unknown variance, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-202-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&lt;</mo><mn>30</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2990" style="width: 3.435em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.82em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.82em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2991"><span class="mi" id="MathJax-Span-2992" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2993" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mn" id="MathJax-Span-2994" style="font-family: STIXGeneral; padding-left: 0.31em;">30</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo><</mo><mn>30</mn></math></span></span><script type="math/tex" id="MathJax-Element-202">n < 30</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-203-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>t</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>&#x2212;</mo><mi>&#x03BC;</mi></mrow><msqrt><mfrac><msup><mi>&#x03C3;</mi><mn>2</mn></msup><mi>n</mi></mfrac></msqrt></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2995" style="width: 5.023em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.101em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1004.1em 4.511em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2996"><span class="mi" id="MathJax-Span-2997" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2998" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2999" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.281em 1002.15em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -1.073em;"><span class="mrow" id="MathJax-Span-3000"><span class="mi" id="MathJax-Span-3001" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-3002" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-3003" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">μ</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.615em 1002.21em 4.767em -999.997em); top: -2.764em; left: 50%; margin-left: -1.073em;"><span class="msqrt" id="MathJax-Span-3004"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.08em 4.511em -999.997em); top: -3.993em; left: 1.078em;"><span class="mrow" id="MathJax-Span-3005"><span class="mfrac" id="MathJax-Span-3006"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.281em 1000.72em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.356em;"><span class="msubsup" id="MathJax-Span-3007"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3008" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.198em; left: 0.412em;"><span class="mn" id="MathJax-Span-3009" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-3010" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.87em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.873em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1001.13em 3.384em -999.997em); top: -4.403em; left: 1.078em;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0.617em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.31em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.308em 1001.08em 4.46em -999.997em); top: -3.686em; left: 0em;"><span style="font-family: STIXSizeOneSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.31em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.308em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.247em; border-left: 0px solid; width: 0px; height: 3.816em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>t</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi>μ</mi></mrow><msqrt><mfrac><msup><mi>σ</mi><mn>2</mn></msup><mi>n</mi></mfrac></msqrt></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-203"> t = \frac{x - \mu}{ \sqrt{\frac{\sigma ^2}{n}}}</script></span></p><h4 id="step-3"><a class="anchor hidden-xs" href="#step-3" title="step-3"><span class="octicon octicon-link"></span></a>Step 3</h4><p>Use tables to find the critical value</p><p>If a <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-204-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3011" style="width: 0.31em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.259em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.847em 1000.26em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3012"><span class="mi" id="MathJax-Span-3013" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.816em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span><script type="math/tex" id="MathJax-Element-204">t</script></span> distribution is used, there are <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-205-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3014" style="width: 2.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3015"><span class="mi" id="MathJax-Span-3016" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-3017" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-3018" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-205">n - 1</script></span> degrees of freedom</p><p>State the critical value, or draw a graph and mark it with the critical value and the test statistic</p><h4 id="step-4"><a class="anchor hidden-xs" href="#step-4" title="step-4"><span class="octicon octicon-link"></span></a>Step 4</h4><p>Conclude the hypothesis test</p><p>As [condition] there is / isn’t significant evidence at the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-206-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">&#x0025;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3019" style="width: 0.976em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3020"><span class="mi" id="MathJax-Span-3021" style="font-family: STIXGeneral;">%</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-206">\%</script></span> level that the mean differs from a.<br>
We therefore reject/accept <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-207-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3022" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-3023"><span class="msubsup" id="MathJax-Span-3024"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3025" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-3026" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-207">H_0</script></span> and conclude that… [context].</p><h3 id="errors"><a class="anchor hidden-xs" href="#errors" title="errors"><span class="octicon octicon-link"></span></a>Errors</h3><h4 id="type-1-error"><a class="anchor hidden-xs" href="#type-1-error" title="type-1-error"><span class="octicon octicon-link"></span></a>Type 1 error</h4><p>This type of error occurs when H<sub>0</sub> is reject, and H<sub>1</sub> is accepted when H<sub>0</sub> is in fact correct.</p><p>The probability of obtaining a type 1 error is the level of significance of the test hypothesis.</p><h4 id="type-2-error"><a class="anchor hidden-xs" href="#type-2-error" title="type-2-error"><span class="octicon octicon-link"></span></a>Type 2 error</h4><p>This type of error occurs when <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-208-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3027" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-3028"><span class="msubsup" id="MathJax-Span-3029"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3030" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-3031" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-208">H_0</script></span> is accepted, when it is in fact false.</p><p>The probability of a type 2 error is not fixed, since it depends upon the extent to which the value of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-209-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3032" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.001em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3033"><span class="mi" id="MathJax-Span-3034" style="font-family: STIXGeneral; font-style: italic;">μ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-209">\mu</script></span> deviates from the value given in <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-210-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3035" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-3036"><span class="msubsup" id="MathJax-Span-3037"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3038" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-3039" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-210">H_0</script></span>. If the value of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-211-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3040" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.001em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3041"><span class="mi" id="MathJax-Span-3042" style="font-family: STIXGeneral; font-style: italic;">μ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-211">\mu</script></span> is closed to the value given in <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-212-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3043" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-3044"><span class="msubsup" id="MathJax-Span-3045"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3046" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-3047" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-212">H_0</script></span> the probability of a type 2 error is large.</p><h2 id="chi-squared-goodness-of-fit-test"><a class="anchor hidden-xs" href="#chi-squared-goodness-of-fit-test" title="chi-squared-goodness-of-fit-test"><span class="octicon octicon-link"></span></a>Chi-squared goodness of fit test</h2><ul>
<li>Used for testing bias, or the independence of variables</li>
<li>The test statistic involves squares, as we are only interested in upper limit critical values</li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-213-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3048" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-3049"><span class="msubsup" id="MathJax-Span-3050"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3051" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-3052" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-213">H_0</script></span> is always that the variables are independent</li>
</ul><h3 id="calculating"><a class="anchor hidden-xs" href="#calculating" title="calculating"><span class="octicon octicon-link"></span></a>Calculating</h3><h4 id="calculating-expected-frequencies"><a class="anchor hidden-xs" href="#calculating-expected-frequencies" title="calculating-expected-frequencies"><span class="octicon octicon-link"></span></a>Calculating expected frequencies</h4><p>For a table of values, the expected frequency of each value is</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-214-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mo>&#x2211;</mo><mi>r</mi><mi>o</mi><mi>w</mi><mo>&#x00D7;</mo><mo>&#x2211;</mo><mi>c</mi><mi>o</mi><mi>l</mi><mi>u</mi><mi>m</mi><mi>n</mi></mrow><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3053" style="width: 10.095em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.251em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.566em 1008.25em 3.076em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-3054"><span class="mfrac" id="MathJax-Span-3055"><span style="display: inline-block; position: relative; width: 8.046em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.076em 1007.89em 4.408em -999.997em); top: -4.711em; left: 50%; margin-left: -3.942em;"><span class="mrow" id="MathJax-Span-3056"><span class="mo" id="MathJax-Span-3057" style="font-family: STIXGeneral; vertical-align: 0.003em;">∑</span><span class="mi" id="MathJax-Span-3058" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-3059" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-3060" style="font-family: STIXGeneral; font-style: italic;">w</span><span class="mo" id="MathJax-Span-3061" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mo" id="MathJax-Span-3062" style="font-family: STIXGeneral; vertical-align: 0.003em; padding-left: 0.259em;">∑</span><span class="mi" id="MathJax-Span-3063" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">c</span><span class="mi" id="MathJax-Span-3064" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-3065" style="font-family: STIXGeneral; font-style: italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-3066" style="font-family: STIXGeneral; font-style: italic;">u</span><span class="mi" id="MathJax-Span-3067" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="mi" id="MathJax-Span-3068" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1001.9em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.92em;"><span class="mrow" id="MathJax-Span-3069"><span class="mi" id="MathJax-Span-3070" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-3071" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-3072" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-3073" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-3074" style="font-family: STIXGeneral; font-style: italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1008.05em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 8.046em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.816em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mo>∑</mo><mi>r</mi><mi>o</mi><mi>w</mi><mo>×</mo><mo>∑</mo><mi>c</mi><mi>o</mi><mi>l</mi><mi>u</mi><mi>m</mi><mi>n</mi></mrow><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-214">\frac{\sum row \times \sum column}{total}</script></span></p><p>If the expected frequency of a particular value is less than <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-215-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3075" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3076"><span class="mn" id="MathJax-Span-3077" style="font-family: STIXGeneral;">5</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-215">5</script></span>, its row or column must be merged.</p><h4 id="the-test-statistic"><a class="anchor hidden-xs" href="#the-test-statistic" title="the-test-statistic"><span class="octicon octicon-link"></span></a>The test statistic</h4><p>For the general case</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-216-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>&#x03C7;</mi><mn>2</mn></msup><mo>=</mo><mo>&#x2211;</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>O</mi><mi>i</mi></msub><mo>&#x2212;</mo><msub><mi>E</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><msub><mi>E</mi><mi>i</mi></msub></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3078" style="width: 10.146em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.302em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1008.3em 3.947em -999.997em); top: -2.969em; left: 0em;"><span class="mrow" id="MathJax-Span-3079"><span class="msubsup" id="MathJax-Span-3080"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3081" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.566em;"><span class="mn" id="MathJax-Span-3082" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3083" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-3084" style="font-family: STIXSizeOneSym; vertical-align: -0.51em; padding-left: 0.31em;">∑</span><span class="mfrac" id="MathJax-Span-3085" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 4.306em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1004.15em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -2.098em;"><span class="mrow" id="MathJax-Span-3086"><span class="mo" id="MathJax-Span-3087" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-3088"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.72em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3089" style="font-family: STIXGeneral; font-style: italic;">O</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mi" id="MathJax-Span-3090" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3091" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-3092" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3093" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-3094" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-3095"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-3096" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.361em;"><span class="mn" id="MathJax-Span-3097" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.87em 4.306em -999.997em); top: -3.327em; left: 50%; margin-left: -0.459em;"><span class="msubsup" id="MathJax-Span-3098"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3099" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-3100" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1004.31em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.306em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.974em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.059em; border-left: 0px solid; width: 0px; height: 3.003em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>χ</mi><mn>2</mn></msup><mo>=</mo><mo>∑</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>O</mi><mi>i</mi></msub><mo>−</mo><msub><mi>E</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><msub><mi>E</mi><mi>i</mi></msub></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-216">\chi ^2 = \sum \frac{(O_i - E_i)^2}{E_i}</script></span></p><p>For the case where the table is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-217-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>&#x00D7;</mo><mn>2</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3101" style="width: 2.666em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3102"><span class="mn" id="MathJax-Span-3103" style="font-family: STIXGeneral;">2</span><span class="mo" id="MathJax-Span-3104" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mn" id="MathJax-Span-3105" style="font-family: STIXGeneral; padding-left: 0.259em;">2</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-217">2\times2</script></span> we have only one degree of freedom, and the test statistic is</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-218-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>&#x03C7;</mi><mn>2</mn></msup><mo>=</mo><mo>&#x2211;</mo><mfrac><mrow><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>O</mi><mi>i</mi></msub><mo>&#x2212;</mo><msub><mi>E</mi><mi>i</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>&#x2212;</mo><mn>0.5</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><msub><mi>E</mi><mi>i</mi></msub></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3106" style="width: 13.886em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.376em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1011.38em 3.947em -999.997em); top: -2.969em; left: 0em;"><span class="mrow" id="MathJax-Span-3107"><span class="msubsup" id="MathJax-Span-3108"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3109" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.566em;"><span class="mn" id="MathJax-Span-3110" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3111" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-3112" style="font-family: STIXSizeOneSym; vertical-align: -0.51em; padding-left: 0.31em;">∑</span><span class="mfrac" id="MathJax-Span-3113" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 7.38em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1007.28em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -3.635em;"><span class="mrow" id="MathJax-Span-3114"><span class="mo" id="MathJax-Span-3115" style="font-family: STIXGeneral;">(</span><span class="texatom" id="MathJax-Span-3116"><span class="mrow" id="MathJax-Span-3117"><span class="mo" id="MathJax-Span-3118" style="font-family: STIXVariants;">|</span></span></span><span class="msubsup" id="MathJax-Span-3119"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.72em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3120" style="font-family: STIXGeneral; font-style: italic;">O</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mi" id="MathJax-Span-3121" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-3122" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-3123" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3124" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-3125" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="texatom" id="MathJax-Span-3126"><span class="mrow" id="MathJax-Span-3127"><span class="mo" id="MathJax-Span-3128" style="font-family: STIXVariants;">|</span></span></span><span class="mo" id="MathJax-Span-3129" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-3130" style="font-family: STIXGeneral; padding-left: 0.259em;">0.5</span><span class="msubsup" id="MathJax-Span-3131"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-3132" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.361em;"><span class="mn" id="MathJax-Span-3133" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.87em 4.306em -999.997em); top: -3.327em; left: 50%; margin-left: -0.459em;"><span class="msubsup" id="MathJax-Span-3134"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-3135" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-3136" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1007.38em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 7.38em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.974em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.059em; border-left: 0px solid; width: 0px; height: 3.003em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>χ</mi><mn>2</mn></msup><mo>=</mo><mo>∑</mo><mfrac><mrow><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>O</mi><mi>i</mi></msub><mo>−</mo><msub><mi>E</mi><mi>i</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>−</mo><mn>0.5</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><msub><mi>E</mi><mi>i</mi></msub></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-218">\chi ^2 = \sum \frac{(|O_i - E_i| - 0.5)^2}{E_i}</script></span></p><h4 id="the-critical-value"><a class="anchor hidden-xs" href="#the-critical-value" title="the-critical-value"><span class="octicon octicon-link"></span></a>The critical value</h4><p>The degrees of freedom, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-219-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BD;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3137" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3138"><span class="mi" id="MathJax-Span-3139" style="font-family: STIXGeneral; font-style: italic;">ν</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ν</mi></math></span></span><script type="math/tex" id="MathJax-Element-219">\nu</script></span> , of a chi-squared contingency table is given by <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-220-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3140" style="width: 2.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3141"><span class="mi" id="MathJax-Span-3142" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-3143" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-3144" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-220">n - 1</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-221-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-3145" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-3146"><span class="mi" id="MathJax-Span-3147" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-221">n</script></span> is the number of groups in the table.</p><p>The critical value is then found from the table or from a calculator.</p><p>Example of the comparison of two variables</p><blockquote>
<p>Natives of England, Africa, and China were classified by blood group</p>
<table>
<thead>
<tr>
<th></th>
<th>O</th>
<th>A</th>
<th>B</th>
<th>AB</th>
</tr>
</thead>
<tbody>
<tr>
<td>English</td>
<td>235</td>
<td>212</td>
<td>79</td>
<td>83</td>
</tr>
<tr>
<td>African</td>
<td>147</td>
<td>106</td>
<td>30</td>
<td>51</td>
</tr>
<tr>
<td>Chinese</td>
<td>162</td>
<td>135</td>
<td>52</td>
<td>43</td>
</tr>
</tbody>
</table>
<p>Is there any evidence at the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-287-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi mathvariant="normal">&#x0025;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4116" style="width: 1.548em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.267em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1001.22em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-4117"><span class="mn" id="MathJax-Span-4118" style="font-family: STIXGeneral;">5</span><span class="mi" id="MathJax-Span-4119" style="font-family: STIXGeneral;">%</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-287">5\%</script></span> level that there is a connection between blood group and nationality?<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-288-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4120" style="width: 1.454em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.173em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.361em 1001.17em 2.438em -999.998em); top: -2.152em; left: 0em;"><span class="mrow" id="MathJax-Span-4121"><span class="msubsup" id="MathJax-Span-4122"><span style="display: inline-block; position: relative; width: 1.173em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.75em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-4123" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.838em; left: 0.705em;"><span class="mn" id="MathJax-Span-4124" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.157em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.226em; border-left: 0px solid; width: 0px; height: 1.089em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-288">H_0</script></span>: There is no connection between blood group and nationality<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-289-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4125" style="width: 1.454em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.173em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.361em 1001.17em 2.438em -999.998em); top: -2.152em; left: 0em;"><span class="mrow" id="MathJax-Span-4126"><span class="msubsup" id="MathJax-Span-4127"><span style="display: inline-block; position: relative; width: 1.173em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.75em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-4128" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.838em; left: 0.705em;"><span class="mn" id="MathJax-Span-4129" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.157em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.226em; border-left: 0px solid; width: 0px; height: 1.089em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-289">H_1</script></span>: There is a connection between blood group and nationality</p>
<p>For each cell we find the expected value by multiplying the row total and column total before dividing by the table total</p>
<table>
<thead>
<tr>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td>24.816</td>
<td>206.65</td>
<td>73.44</td>
<td>80.74</td>
<td></td>
</tr>
<tr>
<td>136.10</td>
<td>113.33</td>
<td>40.28</td>
<td>44.28</td>
<td></td>
</tr>
<tr>
<td>159.74</td>
<td>133.02</td>
<td>47.27</td>
<td>51.97</td>
<td></td>
</tr>
</tbody>
</table>
<p>We have that <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-290-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BD;</mi><mo>=</mo><mo stretchy="false">(</mo><mn>4</mn><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>3</mn><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4130" style="width: 11.337em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.276em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1009.23em 2.859em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-4131"><span class="mi" id="MathJax-Span-4132" style="font-family: STIXGeneral; font-style: italic;">ν</span><span class="mo" id="MathJax-Span-4133" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mo" id="MathJax-Span-4134" style="font-family: STIXGeneral; padding-left: 0.33em;">(</span><span class="mn" id="MathJax-Span-4135" style="font-family: STIXGeneral;">4</span><span class="mo" id="MathJax-Span-4136" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="mn" id="MathJax-Span-4137" style="font-family: STIXGeneral; padding-left: 0.237em;">1</span><span class="mo" id="MathJax-Span-4138" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-4139" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-4140" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-4141" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="mn" id="MathJax-Span-4142" style="font-family: STIXGeneral; padding-left: 0.237em;">1</span><span class="mo" id="MathJax-Span-4143" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-4144" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-4145" style="font-family: STIXGeneral; padding-left: 0.33em;">6</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.283em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ν</mi><mo>=</mo><mo stretchy="false">(</mo><mn>4</mn><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>3</mn><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-290">\nu = (4-1)(3-1) = 6</script></span> degrees of freedom and then that<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-291-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>&#x03C7;</mi><mn>6</mn><mn>2</mn></msubsup><mo stretchy="false">(</mo><mn>5</mn><mi mathvariant="normal">&#x0025;</mi><mo stretchy="false">)</mo><mo>=</mo><mn>12.592</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4146" style="width: 8.527em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.981em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.548em 1006.93em 3em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-4147"><span class="msubsup" id="MathJax-Span-4148"><span style="display: inline-block; position: relative; width: 0.986em; height: 0px;"><span style="position: absolute; clip: rect(3.422em 1000.52em 4.311em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-4149" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.375em 1000.42em 4.124em -999.998em); top: -4.307em; left: 0.564em;"><span class="mn" id="MathJax-Span-4150" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.375em 1000.42em 4.124em -999.998em); top: -3.651em; left: 0.471em;"><span class="mn" id="MathJax-Span-4151" style="font-size: 70.7%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-4152" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-4153" style="font-family: STIXGeneral;">5</span><span class="mi" id="MathJax-Span-4154" style="font-family: STIXGeneral;">%</span><span class="mo" id="MathJax-Span-4155" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-4156" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-4157" style="font-family: STIXGeneral; padding-left: 0.33em;">12.592</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.454em; border-left: 0px solid; width: 0px; height: 1.546em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>χ</mi><mn>6</mn><mn>2</mn></msubsup><mo stretchy="false">(</mo><mn>5</mn><mi mathvariant="normal">%</mi><mo stretchy="false">)</mo><mo>=</mo><mn>12.592</mn></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-291">\chi^2 _6 (5\%) = 12.592</script></span><br>
From the table of expected frequencies<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-292-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>&#x2211;</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>O</mi><mi>i</mi></msub><mo>&#x2212;</mo><msub><mi>E</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><msub><mi>E</mi><mi>i</mi></msub></mfrac><mo>=</mo><mn>8.39</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4158" style="width: 11.15em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.136em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.314em 1009.09em 3.937em -999.998em); top: -2.948em; left: 0em;"><span class="mrow" id="MathJax-Span-4159"><span class="mo" id="MathJax-Span-4160" style="font-family: STIXSizeOneSym; vertical-align: -0.513em;">∑</span><span class="mfrac" id="MathJax-Span-4161" style="padding-left: 0.19em;"><span style="display: inline-block; position: relative; width: 4.265em; height: 0px; margin-right: 0.143em; margin-left: 0.143em;"><span style="position: absolute; clip: rect(3em 1004.12em 4.311em -999.998em); top: -4.635em; left: 50%; margin-left: -2.059em;"><span class="mrow" id="MathJax-Span-4162"><span class="mo" id="MathJax-Span-4163" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-4164"><span style="display: inline-block; position: relative; width: 0.986em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.71em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-4165" style="font-family: STIXGeneral; font-style: italic;">O</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.838em; left: 0.705em;"><span class="mi" id="MathJax-Span-4166" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mo" id="MathJax-Span-4167" style="font-family: STIXGeneral; padding-left: 0.237em;">−</span><span class="msubsup" id="MathJax-Span-4168" style="padding-left: 0.237em;"><span style="display: inline-block; position: relative; width: 0.892em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.66em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-4169" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.838em; left: 0.611em;"><span class="mi" id="MathJax-Span-4170" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-4171"><span style="display: inline-block; position: relative; width: 0.752em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.28em 4.311em -999.998em); top: -3.979em; left: 0em;"><span class="mo" id="MathJax-Span-4172" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.354em; left: 0.33em;"><span class="mn" id="MathJax-Span-4173" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(3.187em 1000.89em 4.265em -999.998em); top: -3.276em; left: 50%; margin-left: -0.419em;"><span class="msubsup" id="MathJax-Span-4174"><span style="display: inline-block; position: relative; width: 0.892em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.66em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-4175" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.838em; left: 0.611em;"><span class="mi" id="MathJax-Span-4176" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; clip: rect(0.892em 1004.26em 1.22em -999.998em); top: -1.309em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.265em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.08em;"></span></span></span></span><span class="mo" id="MathJax-Span-4177" style="font-family: STIXGeneral; padding-left: 0.33em;">=</span><span class="mn" id="MathJax-Span-4178" style="font-family: STIXGeneral; padding-left: 0.33em;">8.39</span></span><span style="display: inline-block; width: 0px; height: 2.953em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.083em; border-left: 0px solid; width: 0px; height: 2.974em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>∑</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>O</mi><mi>i</mi></msub><mo>−</mo><msub><mi>E</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><msub><mi>E</mi><mi>i</mi></msub></mfrac><mo>=</mo><mn>8.39</mn></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-292">\sum \frac{(O_i - E_i)^2}{E_i} = 8.39</script></span></p>
<p>As <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-293-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8.39</mn><mo>&lt;</mo><mn>12.592</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4179" style="width: 7.169em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.857em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1005.81em 2.719em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-4180"><span class="mn" id="MathJax-Span-4181" style="font-family: STIXGeneral;">8.39</span><span class="mo" id="MathJax-Span-4182" style="font-family: STIXGeneral; padding-left: 0.33em;"><</span><span class="mn" id="MathJax-Span-4183" style="font-family: STIXGeneral; padding-left: 0.33em;">12.592</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.111em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8.39</mn><mo><</mo><mn>12.592</mn></math></span></span><script type="math/tex" id="MathJax-Element-293">8.39 < 12.592</script></span> we do not reject <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-294-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4184" style="width: 1.454em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.173em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.361em 1001.17em 2.438em -999.998em); top: -2.152em; left: 0em;"><span class="mrow" id="MathJax-Span-4185"><span class="msubsup" id="MathJax-Span-4186"><span style="display: inline-block; position: relative; width: 1.173em; height: 0px;"><span style="position: absolute; clip: rect(3.187em 1000.75em 4.124em -999.998em); top: -3.979em; left: 0em;"><span class="mi" id="MathJax-Span-4187" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.049em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -3.838em; left: 0.705em;"><span class="mn" id="MathJax-Span-4188" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.157em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.226em; border-left: 0px solid; width: 0px; height: 1.089em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-294">H_0</script></span> at the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-295-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi mathvariant="normal">&#x0025;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4189" style="width: 1.548em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.267em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.689em 1001.22em 2.672em -999.998em); top: -2.527em; left: 0em;"><span class="mrow" id="MathJax-Span-4190"><span class="mn" id="MathJax-Span-4191" style="font-family: STIXGeneral;">5</span><span class="mi" id="MathJax-Span-4192" style="font-family: STIXGeneral;">%</span></span><span style="display: inline-block; width: 0px; height: 2.532em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.054em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-295">5\%</script></span> level and therefore conclude that there is no connection between nationality and blood group.</p>
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var markdown = $(".markdown-body");
//smooth all hash trigger scrolling
function smoothHashScroll() {
var hashElements = $("a[href^='#']").toArray();
for (var i = 0; i < hashElements.length; i++) {
var element = hashElements[i];
var $element = $(element);
var hash = element.hash;
if (hash) {
$element.on('click', function (e) {
// store hash
var hash = this.hash;
if ($(hash).length <= 0) return;
// prevent default anchor click behavior
e.preventDefault();
// animate
$('body, html').stop(true, true).animate({
scrollTop: $(hash).offset().top
}, 100, "linear", function () {
// when done, add hash to url
// (default click behaviour)
window.location.hash = hash;
});
});
}
}
}
smoothHashScroll();
var toc = $('.ui-toc');
var tocAffix = $('.ui-affix-toc');
var tocDropdown = $('.ui-toc-dropdown');
//toc
tocDropdown.click(function (e) {
e.stopPropagation();
});
var enoughForAffixToc = true;
function generateScrollspy() {
$(document.body).scrollspy({
target: ''
});
$(document.body).scrollspy('refresh');
if (enoughForAffixToc) {
toc.hide();
tocAffix.show();
} else {
tocAffix.hide();
toc.show();
}
$(document.body).scroll();
}
function windowResize() {
//toc right
var paddingRight = parseFloat(markdown.css('padding-right'));
var right = ($(window).width() - (markdown.offset().left + markdown.outerWidth() - paddingRight));
toc.css('right', right + 'px');
//affix toc left
var newbool;
var rightMargin = (markdown.parent().outerWidth() - markdown.outerWidth()) / 2;
//for ipad or wider device
if (rightMargin >= 133) {
newbool = true;
var affixLeftMargin = (tocAffix.outerWidth() - tocAffix.width()) / 2;
var left = markdown.offset().left + markdown.outerWidth() - affixLeftMargin;
tocAffix.css('left', left + 'px');
} else {
newbool = false;
}
if (newbool != enoughForAffixToc) {
enoughForAffixToc = newbool;
generateScrollspy();
}
}
$(window).resize(function () {
windowResize();
});
$(document).ready(function () {
windowResize();
generateScrollspy();
});
//remove hash
function removeHash() {
window.location.hash = '';
}
var backtotop = $('.back-to-top');
var gotobottom = $('.go-to-bottom');
backtotop.click(function (e) {
e.preventDefault();
e.stopPropagation();
if (scrollToTop)
scrollToTop();
removeHash();
});
gotobottom.click(function (e) {
e.preventDefault();
e.stopPropagation();
if (scrollToBottom)
scrollToBottom();
removeHash();
});
var toggle = $('.expand-toggle');
var tocExpand = false;
checkExpandToggle();
toggle.click(function (e) {
e.preventDefault();
e.stopPropagation();
tocExpand = !tocExpand;
checkExpandToggle();
})
function checkExpandToggle () {
var toc = $('.ui-toc-dropdown .toc');
var toggle = $('.expand-toggle');
if (!tocExpand) {
toc.removeClass('expand');
toggle.text('Expand all');
} else {
toc.addClass('expand');
toggle.text('Collapse all');
}
}
function scrollToTop() {
$('body, html').stop(true, true).animate({
scrollTop: 0
}, 100, "linear");
}
function scrollToBottom() {
$('body, html').stop(true, true).animate({
scrollTop: $(document.body)[0].scrollHeight
}, 100, "linear");
}
</script>
</body>
</html>