# 超几何分布

参数 概率质量函数 累积分布函数 {\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}}} ${\displaystyle k\,\in \,\left\{\max {(0,\,n+K-N)},\,\dots ,\,\min {(n,\,K)}\right\}}$ ${\displaystyle {{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}}$ ${\displaystyle 1-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}} \over {N \choose K}}\,_{3}F_{2}\!\!\left[{\begin{array}{c}1,\ k+1-K,\ k+1-n\\k+2,\ N+k+2-K-n\end{array}};1\right]}$其中${\displaystyle \,_{p}F_{q}}$为广义超几何函数 ${\displaystyle n{K \over N}}$ ${\displaystyle \left\lceil {\frac {(n+1)(K+1)}{N+2}}\right\rceil -1}$, ${\displaystyle \left\lfloor {\frac {(n+1)(K+1)}{N+2}}\right\rfloor }$ ${\displaystyle n{K \over N}{(N-K) \over N}{N-n \over N-1}}$ ${\displaystyle {\frac {(N-2K)(N-1)^{\frac {1}{2}}(N-2n)}{[nK(N-K)(N-n)]^{\frac {1}{2}}(N-2)}}}$ ${\displaystyle \left.{\frac {1}{nK(N-K)(N-n)(N-2)(N-3)}}\cdot \right.}$ ${\displaystyle {\Big [}(N-1)N^{2}{\Big (}N(N+1)-6K(N-K)-6n(N-n){\Big )}+{}}$ ${\displaystyle {}+6nK(N-K)(N-n)(5N-6){\Big ]}}$ ${\displaystyle {\frac {{N-K \choose n}{_{2}F_{1}(-n,-K;N-K-n+1;e^{t})}}{N \choose n}}}$ ${\displaystyle {\frac {{N-K \choose n}{\,_{2}F_{1}(-n,-K;N-K-n+1;e^{it})}}{N \choose n}}}$

${\displaystyle f(k;n,K,N)={{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}}$