Hypergeometric Distribution approaches Binomial Distribution

Theorem

Let $X$ be a discrete random variable which obeys a hypergeometric distribution over a population consisting of $M$ satisfactory units and $N$ unsatisfactory units:

$\map \Pr {X = r} = \dfrac {\dbinom M r \dbinom N {n - r} } {\dbinom {M + N} r}$

where:

$n$ is the number of trials

$r$ is the number of satisfactory units drawn from the population without replacement.

When $M$ and $N$ are large compared to $n$, $X$ approaches the binomial distribution $\Binomial n p$ where:

$p = \dfrac M {M + N}$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hypergeometric distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hypergeometric distribution