$X$ is a hypergeometric random variable with parameters $M, K, n$ if

$f_X(x) = \frac{ { K \choose x} {M-K \choose n-x} }{ {M \choose n} }$ , $x=\{0,1,...,n\}$

Parameters:

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$M \in \{1,2,...\}$

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$K \in \{0,1,...,M\}$

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$n \in \{1,2,...,M\}$

Syntax:

$X\sim Hypergeo(M,K,n)$

Notes:

1. $X$ represents the number of ``special'' items (from the $K$ special items) present on a sample of size $n$ from a population with $M$ items.
2. The expected value of $X$ is noted as $E[X] = n \frac{K}{M}$
3. The variance of $X$ is noted as $Var[X] = n \frac{K}{M} \frac{M-K}{M} \frac{M-n}{M-1}$

Approximation techniques:

If ${K \choose 2} << n, M-K+1-n$ then $X$ can be approximated as a binomial random variable with parameters $n=K$ and $p=\frac{M-K+1-n}{M-K+1}$ . This approximation simplifies the distribution by looking at a system with replacement for large values of $M$ and $K$ .