Referring to this problem: http://karlrosaen.com/ml/hw/all-of-statistics-chapter-2-independence-of-rvs-associated-with-coin-flips/
must have flipped the coin at least x times in order to have yielded x heads? I can buy it, but I'm still a tad frustrated that I don't think I could have come up with this myself.
I was confused by this too but can explain it. Iterating from n=0 is actually correct! It's just that the first x terms of the summation are all 0, so it ends up being equal to starting at x. They're 0 for the reason you give, you need at least x flips to get x heads. The only reason to make starting at × explicit is the later substitution trick with m I think.
I think this also leads to an easier way to understand the showing the independence part instead of thinking about indicator functions. If you just use the formula for marginal probability as originally given without the indicator it is correct -but the only nonzero term in the sum is the one where n=x+y.
Referring to this problem: http://karlrosaen.com/ml/hw/all-of-statistics-chapter-2-independence-of-rvs-associated-with-coin-flips/
I was confused by this too but can explain it. Iterating from n=0 is actually correct! It's just that the first x terms of the summation are all 0, so it ends up being equal to starting at x. They're 0 for the reason you give, you need at least x flips to get x heads. The only reason to make starting at × explicit is the later substitution trick with m I think.
I think this also leads to an easier way to understand the showing the independence part instead of thinking about indicator functions. If you just use the formula for marginal probability as originally given without the indicator it is correct -but the only nonzero term in the sum is the one where n=x+y.