Homepage updates.

Author: localhost433

index.html

@@ -31,11 +31,11 @@
         <article>
             <h1>Hello, I'm Robin.</h1>
             <p>
-                A little bit of self intro, I'm a <del>freshman</del> sophomore at the NYU studying Computer Science and Mathematics.
-                This site serves as a platform for me to document my thoughts, projects, and anything else I find
-                worth sharing. I share random ideas and stuff I created about school, life, etc. (Not any type of
-                category that I want to fall into, journal, diary, because it would feel kind of restrictive on
-                'how'/'what' should I write about anything.)
+                A little bit of self intro, I'm a <del>freshman</del> sophomore at the NYU studying Mathematics and
+                Computer Science. (Planning on getting a minor in linguistics) This site serves as a platform for me
+                to document my thoughts, projects, and anything else I find worth sharing. I share random ideas and
+                stuff I created about school, life, etc. (Not any type of category that I want to fall into, journal,
+                diary, because it would feel kind of restrictive on 'how'/'what' should I write about anything.)
             </p>
 
             <h2>What You Might Find Here</h2>
@@ -74,8 +74,7 @@ <h2>Connect with Me</h2>
             <h2>Tech Stack</h2>
             <p>
                 This site is built with vanilla HTML, CSS, and JavaScript. I'm trying to use minimal dependencies to
-                stay
-                lightweight, the ones being used including:
+                stay lightweight, the ones being used including:
             </p>
             <ul>
                 <li><code>js-yaml</code> - parse front-matter in notes</li>
@@ -90,17 +89,17 @@ <h2>Tech Stack</h2>
 
             <h2>Future Plans</h2>
             <p>
-                I plan to add more features and improve the design over time. Some ideas include:
+                I plan to add more features and improve the design over time. Email me if you have suggestions!
+                Currently, some ideas include:
             </p>
             <ul>
                 <li><del>Improving the comment system</del> (Done)</li>
-                <li>Converting all my course notes into markdown</li>
-                <li>Organizing stuff I wrote during the summer into blog posts</li>
+                <li><u>Converting all my course notes into markdown</u> (Done partial, in progress)</li>
+                <li><u>Organizing stuff I wrote during the summer into blog posts</u> (In progress)</li>
                 <li>Adding a search feature for easier navigation</li>
-                <li><u>Enhancing the overall design and responsiveness</u> (In progress)</li>
+                <li><del>Enhance the overall design and responsiveness</del> (Done)</li>
                 <li><u>Supporting more Markdown features</u> (In progress for parsing in general)</li>
-                <li><u>Notifying users of new posts or updates</u> (In progress)</li>
-                <li>Add a section for ideas pop up in my mind</li>
+                <li>Notifying users of new posts or updates</li>
         </article>
     </main>
 

notes/courses/MATH-UA-333/04-05-drv.md

@@ -4,52 +4,65 @@ date: 2025-09-16/18/23/25
 ---
 
 ## Expectation
+
 For a discrete random variable $X$ with p.m.f. $p_X(t)=P[X=t]$,
 $$
 \E[X] = \sum_{t} t p_X(t), \quad \E[f(X)] = \sum_t f(t) p_X(t),
 $$
 
 **Interpretations**
+
 - Weighted average over the values of $X$: $\sum_t t P[X=t]$.
 - Weighted average over outcomes $\omega$ in the sample space $S$: $\sum_{\omega\in S} X(\omega) P(\{\omega\})$.
 
 > Examples:
+> 
 > - Fair die $X\in\{1,\dots,6\}$: $\E[X]=3.5$.
 > - Heads in three fair flips $Y\in\{0,1,2,3\}$: $\E[Y]=1.5$.
 > - Indicator ${\mathbf{1}}_A$: $\E[{\mathbf{1}}_A]=P(A)$.
 
 ## Linearity of expectation
+
 For random variables $X,Y$ and scalar $a\in\mathbb{R}$,
 $$
 \E[aX+Y] = a \E[X] + \E[Y].
 $$
 No independence required.
 
 **Counting-by-indicators trick**
+
 If $X=\sum_{i=1}^n {\mathbf{1}}_{A_i}$ counts how many events $A_i$ occur, then
+
 $$
 \E[X] = \sum_{i=1}^n P(A_i).
 $$
+
 > Example: number of fixed points in a random permutation - $\E[X]=1$.
 
 ## Variance
+
 Variance measures spread around the mean:
+
 $$
 \operatorname{Var}(X) = \E[(X-\E X)^2] = \E[X^2] - (\E X)^2.
 $$
+
 Useful identities:
+
 - For constants $a,b$: $\operatorname{Var}(aX+b)=a^2 \operatorname{Var}(X)$.
 - For an indicator: $\operatorname{Var}({\mathbf{1}}_A)=P(A)\big(1-P(A)\big)$.
 - If $X=\sum_{i=1}^n {\mathbf{1}}_{A_i}$ with independent indicators, then
-  $$
-  \operatorname{Var}(X)=\sum_{i=1}^n P(A_i)\big(1-P(A_i)\big).
-  $$
+  
+$$
+\operatorname{Var}(X)=\sum_{i=1}^n P(A_i)\big(1-P(A_i)\big).
+$$
 
 ## Inclusion–Exclusion via indicators
-Using ${\mathbf{1}}_{A^c} = 1 - {\mathbf{1}}_A$ and expanding
-${\mathbf{1}}_{\cup_i E_i} = 1 - \prod_{i=1}^n (1-{\mathbf{1}}_{E_i})$, then taking expectations yields
+
+Using ${\mathbf{1}}_{A^c} = 1 - {\mathbf{1}}_A$ and expanding ${\mathbf{1}}_{\cup_i E_i} = 1 - \prod_{i=1}^n (1-{\mathbf{1}}_{E_i})$, then taking expectations yields
+  
 $$
-P\Big( \bigcup_{i=1}^n\ E_i \Big)
+P\Big( \bigcup_{i=1}^n\ {\E}_i \Big)
 = \sum_{k=1}^n (-1)^{k-1}
 \sum_{1 \leq i_1 < \cdots < i_k \leq n}
 P(\E_{i_1} \cap \cdots \cap \E_{i_k}).
@@ -61,7 +74,8 @@ If $X\in\{0,1\}$ takes value $1$ with probability $p$, we write $X\sim\operatorn
 
 ### Binomial
 
-Let $X$ be the total number of successes in $n$ independent Bernoulli$(p)$ trials.  Then $X\sim \operatorname{Bin}(n,p)$ and can be written $X=X_1+\cdots+X_n$ with $X_i\sim\operatorname{Ber}(p)$.  The pmf is
+Let $X$ be the total number of successes in $n$ independent Bernoulli$(p)$ trials.  Then $X\sim \operatorname{Bin}(n,p)$ and can be written $X=X_1+\cdots+X_n$ with $X_i\sim\operatorname{Ber}(p)$. The pmf is
+
 $$
 P(X=k) = {n\choose k} p^k (1-p)^{ n-k}.
 $$

posts/metadata/entries.json

@@ -68,7 +68,7 @@
         "author": "R",
         "location": "Futian district, Shenzhen, Guangdong, China",
         "tags": ["cs"],
-        "slug": "014-server-dev"
+        "slug": "014-server"
     }
     ,
     {
@@ -77,6 +77,6 @@
         "author": "R",
         "location": "Warren Weaver Hall, NYU, New York, NY",
         "tags": ["notes", "cs"],
-        "slug": "015-seminar-anubis"
+        "slug": "020-seminar-anubis"
     }
 ]