超幾何分布
例えば、赤い玉10個と白い玉20個を混ぜた、計30個の玉を入れた壺の中から5個の球を取り出すとき、赤い玉がちょうど1つである確率は
( 10 1 ) ( 30 − 10 5 − 1 ) ( 30 5 ) = 8075 23751 ≈ 0.34 {\displaystyle {\frac {{\binom {10}{1}}{\binom {30-10}{5-1}}}{\binom {30}{5}}}={\frac {8075}{23751}}\approx 0.34}
赤い玉の個数の期待値は
5 × 10 30 ≈ 1.67 {\displaystyle {\frac {5\times 10}{30}}\approx 1.67}
期待値 E ( X ) = n ⋅ K N {\displaystyle E(X)=n\cdot {\frac {K}{N}}}
分散 Var  ( X ) = n ⋅ K N ⋅ N − K N ⋅ N − n N − 1 {\displaystyle \operatorname {Var} (X)=n\cdot {\frac {K}{N}}\cdot {\frac {N-K}{N}}\cdot {\frac {N-n}{N-1}}}
最頻値 ⌊ ( n + 1 ) ( K + 1 ) N + 2 ⌋ {\displaystyle \left\lfloor {\frac {(n+1)(K+1)}{N+2}}\right\rfloor }
対称性 f X ( k ; N , K , n ) = f X ( k ; N , n , K ) = f X ( n − k ; N , N − K , n ) = f X ( K − k ; N , K , N − n ) {\displaystyle {\begin{aligned}f_{X}(k;N,K,n)&=f_{X}(k;N,n,K)\\&=f_{X}(n-k;N,N-K,n)\\&=f_{X}(K-k;N,K,N-n)\end{aligned}}}