$X$ is a negative hypergeometric random variable with parameters $W, B, b$ if

$f_X(x) = \frac{ { x+b-1 \choose x} {W+B-b-x \choose W-x} }{ {W+B \choose W} }$ , $x=\{0,1,...,W\}$

Parameters:

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

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

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$b \in \{1,2,...,B\}$

Syntax:

$X\sim NegHypergeo(W,B,b)$

Notes:

1. $X$ represents the number of ``special'' items (from the $W$ special items) present before the $b$ th object from a population with $B$ items.
2. The expected value of $X$ is noted as $E[X] = \frac{Wb}{B+1}$
3. The variance of $X$ is noted as $Var[X] = \frac{Wb(B-b+1)(W+B+1)}{(B+2)(B+1)^2}$

Approximation techniques:

If ${x \choose 2} << W$ and ${b \choose 2} << B$ then $X$ can be approximated as a negative binomial random variable with parameters $r = b$ and $p = \frac{W}{W+B}$ . This approximation simplifies the distribution by looking at a system with replacement for large values of $W$ and $B$ .