| Hypergeometric distribution | CDF | 1 − ( n k + 1 ) ( N − n K − k − 1 ) ( N K ) 3 F 2 [ 1 , k + 1 − K , k + 1 − n k + 2 , N + k + 2 − K − n ; 1 ] , {\displaystyle 1-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}} \ove… | 1.00 | infobox |
| Hypergeometric distribution | CF | ( N − K n ) 2 F 1 ( − n , − K ; N − K − n + 1 ; e i t ) ( N n ) {\displaystyle {\frac {{\binom {N-K}{n}}\,_{2}F_{1}(-n,-K\,;\,N-K-n+1\,;\,e^{it})}{\binom {N}{n}}}} | 1.00 | infobox |
| Hypergeometric distribution | Excess kurtosis | 1 n K ( N − K ) ( N − n ) ( N − 2 ) ( N − 3 ) ⋅ {\displaystyle \left.{\frac {1}{nK(N-K)(N-n)(N-2)(N-3)}}\cdot \right.} [ ( N − 1 ) N 2 ( N ( N + 1 ) − 6 K ( N − K ) − 6 n ( N −… | 1.00 | infobox |
| Hypergeometric distribution | Mean | n K N {\displaystyle n{K \over N}} | 1.00 | infobox |
| Hypergeometric distribution | Median | ⌊ n K N ⌋ {\displaystyle \left\lfloor n{K \over N}\right\rfloor } or ⌈ n K N ⌉ {\displaystyle \left\lceil n{K \over N}\right\rceil } | 1.00 | infobox |
| Hypergeometric distribution | MGF | ( N − K n ) 2 F 1 ( − n , − K ; N − K − n + 1 ; e t ) ( N n ) {\displaystyle {\frac {{\binom {N-K}{n}}\,_{2}F_{1}(-n,-K\,;\,N-K-n+1\,;\,e^{t})}{\binom {N}{n}}}} | 1.00 | infobox |
| Hypergeometric distribution | Mode | ⌈ ( n + 1 ) ( K + 1 ) N + 2 ⌉ − 1 , ⌊ ( n + 1 ) ( K + 1 ) N + 2 ⌋ {\displaystyle \left\lceil {\frac {(n+1)(K+1)}{N+2}}\right\rceil -1,\left\lfloor {\frac {(n+1)(K+1)}{N+2}}\righ… | 1.00 | infobox |
| Hypergeometric distribution | Notation | H y p e r g e o m e t r i c ( N , K , n ) {\displaystyle \mathrm {Hypergeometric} (N,K,n)} | 1.00 | infobox |
| Hypergeometric distribution | Parameters | N ∈ { 0 , 1 , 2 , … } K ∈ { 0 , 1 , 2 , … , N } n ∈ { 0 , 1 , 2 , … , N } {\displaystyle {\begin{aligned}N&\in \left\{0,1,2,\dots \right\}\\K&\in \left\{0,1,2,\dots ,N\right\}\\… | 1.00 | infobox |
| Hypergeometric distribution | PMF | ( K k ) ( N − K n − k ) ( N n ) {\displaystyle {\frac {{\binom {K}{k}}{\binom {N-K}{n-k}}}{\binom {N}{n}}}} | 1.00 | infobox |
| Hypergeometric distribution | Skewness | ( N − 2 K ) ( N − 1 ) 1 2 ( N − 2 n ) [ n K ( N − K ) ( N − n ) ] 1 2 ( N − 2 ) {\displaystyle {\frac {(N-2K)(N-1)^{\frac {1}{2}}(N-2n)}{[nK(N-K)(N-n)]^{\frac {1}{2}}(N-2)}}} | 1.00 | infobox |
| Hypergeometric distribution | Support | k ∈ { max ( 0 , n + K − N ) , … , min ( n , K ) } {\displaystyle \scriptstyle {k\,\in \,\{\max {(0,\,n+K-N)},\,\dots ,\,\min {(n,\,K)}\}}\,} | 1.00 | infobox |
| Hypergeometric distribution | Variance | n K N N − K N N − n N − 1 {\displaystyle n{K \over N}{N-K \over N}{N-n \over N-1}} | 1.00 | infobox |
| Hypergeometric distribution | is a | discrete probability distribution that describes the probability of k | 0.90 | text |
