Hypergeometric distribution

Hypergeometric

Parameters	${\displaystyle {\begin{aligned}N&\in \left\{0,1,2,\dots \right\}\\K&\in \left\{0,1,2,\dots ,N\right\}\\n&\in \left\{0,1,2,\dots ,N\right\}\end{aligned}}\,}$
Support	${\displaystyle \scriptstyle {k\,\in \,\left\{\max {(0,\,n+K-N)},\,\dots ,\,\min {(n,\,K)}\right\}}\,}$
pmf	${\displaystyle {{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}}$
CDF	${\displaystyle 1-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}} \over {N \choose K}}\,_{3}F_{2}\!\!\left[{\begin{array}{c}1,\ k+1-K,\ k+1-n\\k+2,\ N+k+2-K-n\end{array}};1\right],}$ where ${\displaystyle \,_{p}F_{q}}$ is the generalized hypergeometric function
Mean	${\displaystyle n{K \over N}}$
Mode	${\displaystyle \left\lfloor {\frac {(n+1)(K+1)}{N+2}}\right\rfloor }$
Variance	${\displaystyle n{K \over N}{(N-K) \over N}{N-n \over N-1}}$
Skewness	${\displaystyle {\frac {(N-2K)(N-1)^{\frac {1}{2}}(N-2n)}{[nK(N-K)(N-n)]^{\frac {1}{2}}(N-2)}}}$
Ex. kurtosis	${\displaystyle \left.{\frac {1}{nK(N-K)(N-n)(N-2)(N-3)}}\cdot \right.}$ ${\displaystyle {\Big [}(N-1)N^{2}{\Big (}N(N+1)-6K(N-K)-6n(N-n){\Big )}+}$ ${\displaystyle 6nK(N-K)(N-n)(5N-6){\Big ]}}$
MGF	${\displaystyle {\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{t})}}{N \choose n}}\,\!}$
CF	${\displaystyle {\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{it})}}{N \choose n}}}$

Multivariate Hypergeometric Distribution

Parameters	${\displaystyle c\in \mathbb {N} =\lbrace 0,1,\ldots \rbrace }$ ${\displaystyle (K_{1},\ldots ,K_{c})\in \mathbb {N} ^{c}}$ ${\displaystyle N=\sum _{i=1}^{c}K_{i}}$ ${\displaystyle n\in \lbrace 0,\ldots ,N\rbrace }$
Support	${\displaystyle \left\{\mathbf {k} \in \mathbb {Z} _{0+}^{c}\,:\,\forall i\ k_{i}\leq K_{i},\sum _{i=1}^{c}k_{i}=n\right\}}$
pmf	${\displaystyle {\frac {\prod _{i=1}^{c}{\binom {K_{i}}{k_{i}}}}{\binom {N}{n}}}}$
Mean	${\displaystyle E(X_{i})={\frac {nK_{i}}{N}}}$
Variance	${\displaystyle {\text{Var}}(X_{i})={\frac {K_{i}}{N}}\left(1-{\frac {K_{i}}{N}}\right)n{\frac {N-n}{N-1}}}$ ${\displaystyle {\text{Cov}}(X_{i},X_{j})=-{\frac {nK_{i}K_{j}}{N^{2}}}{\frac {N-n}{N-1}}}$

${\displaystyle \left.{\frac {1}{nK(N-K)(N-n)(N-2)(N-3)}}\cdot \right.}$ ${\displaystyle {\Big [}(N-1)N^{2}{\Big (}N(N+1)-6K(N-K)-6n(N-n){\Big )}+}$

In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of ${\displaystyle k}$ successes in ${\displaystyle n}$ draws, without replacement, from a finite population of size ${\displaystyle N}$ that contains exactly ${\displaystyle K}$ successes, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of ${\displaystyle k}$ successes in ${\displaystyle n}$ draws with replacement.

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