Hypergeometric distribution

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	$\scriptstyle {k\,\in \,\left\{\max {(0,\,n+K-N)},\,\dots ,\,\min {(n,\,K)}\right\}}\,$
PMF	${{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}$
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	${\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{t})}}{N \choose n}}\,\!$
CF	${\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{it})}}{N \choose n}}$