超几何分布¶
带参数的超几何随机变量 \(\left(M,n,N\right)\) 计算样本大小中“好”对象的数量 \(N\) 从一个没有替换的人口中选择 \(M\) 其中的对象 \(n\) 是总人口中“好”对象的数量。
\BEGIN{eqnarray *}} p\left(k;N,n,M\right) & = & \frac{{\left(\begin{{array}}{{c}} n\\ k\end{{array}}\right)\left(\begin{{array}}{{c}} M-n\\ N-k\end{{array}}\right)}}{{\left(\begin{{array}}{{c}} M\\ N\end{{array}}\right)}}\quad N-\left(M-n\right)\leq k\leq\min\left(n,N\right)\\ F\left(x;N,n,M\right) & = & \sum_{{k=0}}^{{\left\lfloor x\right\rfloor }}\frac{{\left(\begin{{array}}{{c}} m\\ k\end{{array}}\right)\left(\begin{{array}}{{c}} N-m\\ n-k\end{{array}}\right)}}{{\left(\begin{{array}}{{c}} N\\ n\end{{array}}\right)}},\\ \mu & = & \frac{{nN}}{{M}}\\ \mu_{{2}} & = & \frac{{nN\left(M-n\right)\left(M-N\right)}}{{M^{{2}}\left(M-1\right)}}\\ \gamma_{{1}} & = & \frac{{\left(M-2n\right)\left(M-2N\right)}}{{M-2}}\sqrt{{\frac{{M-1}}{{nN\left(M-m\right)\left(M-n\right)}}}}\\ \gamma_{{2}} & = & \frac{{g\left(N,n,M\right)}}{{nN\left(M-n\right)\left(M-3\right)\left(M-2\right)\left(N-M\right)}}\end{{eqnarray* }
位置(定义 \(m=M-n\) )
\BEGIN{eqnarray *}} g\left(N,n,M\right) & = & m^{{3}}-m^{{5}}+3m^{{2}}n-6m^{{3}}n+m^{{4}}n+3mn^{{2}}\\ & & -12m^{{2}}n^{{2}}+8m^{{3}}n^{{2}}+n^{{3}}-6mn^{{3}}+8m^{{2}}n^{{3}}\\ & & +mn^{{4}}-n^{{5}}-6m^{{3}}N+6m^{{4}}N+18m^{{2}}nN\\ & & -6m^{{3}}nN+18mn^{{2}}N-24m^{{2}}n^{{2}}N-6n^{{3}}N\\ & & -6mn^{{3}}N+6n^{{4}}N+6m^{{2}}N^{{2}}-6m^{{3}}N^{{2}}-24mnN^{{2}}\\ & & +12m^{{2}}nN^{{2}}+6n^{{2}}N^{{2}}+12mn^{{2}}N^{{2}}-6n^{{3}}N^{{2}}.\end{{eqnarray* }