Show that the hypergeometric probability function approaches...

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.

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.

Key Concepts

Convergence of Distributions

This concept involves one probability distribution approaching another under certain limiting conditions. Specifically, it explains how the hypergeometric distribution, under the condition of a large population and constant success probability, converges to the binomial distribution, illustrating how dependency introduced by sampling without replacement can vanish as the sample size becomes negligible relative to the population.

Asymptotic Analysis

This is a method used to describe the behavior of functions or sequences as an argument or parameter tends to a limit, often infinity. In this context, it involves analyzing how the hypergeometric distribution behaves and simplifies to the binomial distribution when the population size grows very large.

Binomial Distribution

This distribution describes the probability of obtaining a given number of successes in a fixed number of independent trials with replacement (or with independent outcomes) when the probability of success remains constant throughout the trials. It is typically applied to scenarios where the outcomes are independent and each trial has two possible outcomes.

Hypergeometric Distribution

This distribution models the probability of obtaining a certain number of successes in draws without replacement from a finite population that contains a fixed number of successes. It is used when the sampling is done without replacement, which means that each draw reduces the population size and alters the probabilities for subsequent draws.

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