Hypergeometric distribution

Hypergeometric
Hypergeometric
	Probability mass function
	Cumulative distribution function
Parameters
Support
PMF
CDF	where is the generalized hypergeometric function
Mean
Mode
Variance
Skewness
Excess kurtosis	; ;
MGF
CF

In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of $k$ successes (random draws for which the object drawn has a specified feature) in $n$ draws, without replacement, from a finite population of size $N$ that contains exactly $K$ objects with that feature, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of $k$ successes in $n$ draws with replacement.

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