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I was following the proof listed at "Proof that the hypergeometric distribution with large $N$ approaches the binomial distribution." and I can't follow how Sasha goes from their third line of working to their fourth. I'd indent it here but I'm not good at typesetting and don't want to make an ugly mess.

Community
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Tom Roth
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  • Did you leave a comment to @Sasha on the other page? – Did May 07 '14 at 06:59
  • I didn't want to leave a comment for something done more than a year ago. I'm new here and not sure on etiquette. – Tom Roth May 07 '14 at 09:56
  • If you are putting a user's proof under scrutiny, signaling to this user that you are doing so seems only natural, right? – Did May 07 '14 at 11:32
  • Other discussion groups out there feel aggrieved by the resurrecting of an old thread - point taken, however. – Tom Roth May 07 '14 at 12:07
  • Did you leave a comment on the other page, now? – Did May 07 '14 at 20:53

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The original said $$\begin{eqnarray} \frac{\binom{r}{x} \binom{N-r}{n-x}}{\binom{N}{n}} &=& \frac{r!}{\color\green{x!} \cdot (r-x)!} \frac{(N-r)!}{\color\green{(n-x)!} \cdot (N-n -(r-x))!} \cdot \frac{\color\green{n!} \cdot (N-n)!}{N!} \\ &=& \color\green{\binom{n}{x}} \cdot \frac{r!/(r-x)!}{N!/(N-x)!} \cdot \frac{(N-r)! \cdot (N-n)!}{(N-x)! \cdot (N-r-(n-x))!} \\ &=& \binom{n}{x} \cdot \frac{r!/(r-x)!}{N!/(N-x)!} \cdot \frac{(N-r)!/(N-r-(n-x))!}{(N-n+(n-x))!/(N-n)! } \\ &=& \binom{n}{x} \cdot \prod_{k=1}^x \frac{(r-x+k)}{(N-x+k)} \cdot \prod_{m=1}^{n-x}\frac{(N-r-(n-x)+m)}{(N-n+m) } \end{eqnarray}$$

To take one example $$ {r!/(r-x)!} = r \times(r-1) \times \cdots \times(r-x+1) = \prod_{k=0}^{x-1} (r-k)$$ and repeat for the other three cases. Somehow this has become $\displaystyle \prod_{k=1}^{x} (r-k)$ but it makes no difference for the limit.

Henry
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