negative hypergeometric random variable

$X$ 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:

• $\star$

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

• $\star$

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

• $\star$

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

Syntax:

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

Notes:

1. 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. 2.

The expected value of $X$ is noted as $E[X]=\frac{Wb}{B+1}$

3. 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}< and ${b\choose 2}< 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 $B$.

Title negative hypergeometric random variable NegativeHypergeometricRandomVariable 2013-03-22 12:25:05 2013-03-22 12:25:05 alozano (2414) alozano (2414) 16 alozano (2414) Definition msc 62E15 negative hypergeometric distribution