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.
$X$ 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 $B$ .

单词	NegativeHypergeometricRandomVariable
释义	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. $X$ 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 $B$ .
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