negative hypergeometric random variable
$X$ is a negative hypergeometric random variable with parameters $W,B,b$ if
${f}_{X}(x)=\frac{\left(\genfrac{}{}{0pt}{}{x+b1}{x}\right)\left(\genfrac{}{}{0pt}{}{W+Bbx}{Wx}\right)}{\left(\genfrac{}{}{0pt}{}{W+B}{W}\right)}$, $x=\{0,1,\mathrm{\dots},W\}$
Parameters:

$\star $
$W\in \{1,2,\mathrm{\dots}\}$

$\star $
$B\in \{1,2,\mathrm{\dots}\}$

$\star $
$b\in \{1,2,\mathrm{\dots},B\}$
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 $Var[X]=\frac{Wb(Bb+1)(W+B+1)}{(B+2){(B+1)}^{2}}$
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  20130322 12:25:05 
Last modified on  20130322 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 