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Show that the hypergeometric probability function
Chapter 3, Problem 212SE(choose chapter or problem)
Show that the hypergeometric probability function approaches the binomial in the limit as \(N \rightarrow \infty\) and \(p=r / N\) remains constant. That is, show that
\(\lim _{N \rightarrow \infty} \frac{\left(\begin{array}{l} r \\ y \end{array}\right)\left(\begin{array}{l} N-r \\ n-y \end{array}\right)}{\left(\begin{array}{l} N \\ n \end{array}\right)}=\left(\begin{array}{l} n \\ y \end{array}\right) p^{y} q^{n-y}\)
for \(p=r / N\) constant.
Questions & Answers
QUESTION:
Show that the hypergeometric probability function approaches the binomial in the limit as \(N \rightarrow \infty\) and \(p=r / N\) remains constant. That is, show that
\(\lim _{N \rightarrow \infty} \frac{\left(\begin{array}{l} r \\ y \end{array}\right)\left(\begin{array}{l} N-r \\ n-y \end{array}\right)}{\left(\begin{array}{l} N \\ n \end{array}\right)}=\left(\begin{array}{l} n \\ y \end{array}\right) p^{y} q^{n-y}\)
for \(p=r / N\) constant.
ANSWER:Step 1 of 4
Here we have to show that the Hypergeometric probability function approaches the Binomial probability function as \(N \rightarrow \infty\) and \(\mathrm{p}=\frac{r}{N}\).
That is we have to show that
\(\lim _{N \rightarrow \infty} \frac{\left(\begin{array}{l} r \\ y \end{array}\right)\left(\begin{array}{c} N-r \\ n-y \end{array}\right)}{\left(\begin{array}{c} N \\ n \end{array}\right)}=\left(\begin{array}{l} n \\ y \end{array}\right) p^{y} q^{n-y}\)