hypergeometric random variable

$X$ is a hypergeometric random variable with parameters $M, K, n$ if

$f_{X}(x)=\\frac{{K\\choose x}{M-K\\choose n-x}}{{M\\choose n}}$ , $x=\\{0,1,...,n\\}$

Parameters:

$\\star$
$M\\in\\{1,2,...\\}$
$\\star$
$K\\in\\{0,1,...,M\\}$
$\\star$
$n\\in\\{1,2,...,M\\}$

Syntax:

$X\\sim Hypergeo(M,K,n)$

Notes:

1.
$X$ represents the number of “special” items (from the $K$ special items) present on a sample of $n$ from a population with $M$ items.
2.
The expected value of $X$ is noted as $E[X]=n\\frac{K}{M}$
3.
The variance of $X$ is noted as $Var[X]=n\\frac{K}{M}\\frac{M-K}{M}\\frac{M-n}{M-1}$

Approximation techniques:

If ${K\\choose 2}<<n,M-K+1-n$ then $X$ can be approximated as a binomial random variable with parameters $n=K$ and $p=\\frac{M-K+1-n}{M-K+1}$ .This approximation simplifies the distribution by looking at a system with replacement for large values of $M$ and $K$ .

单词	HypergeometricRandomVariable
释义	hypergeometric random variable $X$ is a hypergeometric random variable with parameters $M, K, n$ if $f_{X}(x)=\\frac{{K\\choose x}{M-K\\choose n-x}}{{M\\choose n}}$ , $x=\\{0,1,...,n\\}$ Parameters: $\\star$ $M\\in\\{1,2,...\\}$ $\\star$ $K\\in\\{0,1,...,M\\}$ $\\star$ $n\\in\\{1,2,...,M\\}$ Syntax: $X\\sim Hypergeo(M,K,n)$ Notes: 1. $X$ represents the number of “special” items (from the $K$ special items) present on a sample of $n$ from a population with $M$ items. 2. The expected value of $X$ is noted as $E[X]=n\\frac{K}{M}$ 3. The variance of $X$ is noted as $Var[X]=n\\frac{K}{M}\\frac{M-K}{M}\\frac{M-n}{M-1}$ Approximation techniques: If ${K\\choose 2}<<n,M-K+1-n$ then $X$ can be approximated as a binomial random variable with parameters $n=K$ and $p=\\frac{M-K+1-n}{M-K+1}$ .This approximation simplifies the distribution by looking at a system with replacement for large values of $M$ and $K$ .
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