negative hypergeometric random variable

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

$f_X(x)=\frac{\left(\genfrac{}{}{0pt}{}{x+b-1}{x}\right)\left(\genfrac{}{}{0pt}{}{W+B-b-x}{W-x}\right)}{\left(\genfrac{}{}{0pt}{}{W+B}{W}\right)}$, $x=\{0,1,\mathrm{\dots },W\}$

Parameters:

• $W\in \{1,2,\mathrm{\dots }\}$

• $B\in \{1,2,\mathrm{\dots }\}$

• $b\in \{1,2,\mathrm{\dots },B\}$

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 and 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
Canonical name	NegativeHypergeometricRandomVariable
Date of creation	2013-03-22 12:25:05
Last modified on	2013-03-22 12:25:05
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	16
Author	alozano (2414)
Entry type	Definition
Classification	msc 62E15
Synonym	negative hypergeometric distribution