超幾何分布の性質

超幾何分布において,\(N \rightarrow \infty,~\frac{M}{N} \rightarrow p\)とすると,\(2\)項分布\(Bin(K,p)\)に収束する.\[\lim_{N \rightarrow \infty}P(X=x | N,M,K)=\left(\begin{array}{c} K \\ x \\ \end{array} \right)p^{x}(1-p)^{K-x}\]
証明
\[
\begin{align}
&P(X=x | N,M,K)\\
=&\frac{\left(\begin{array}{c} M \\ x \\ \end{array} \right)\left(\begin{array}{c} N-M \\ K-x \\ \end{array} \right)}{\left(\begin{array}{c} N \\ K \\ \end{array} \right)}\\
=&\frac{\frac{M!}{x!(M-x)!}\frac{(N-M)!}{(K-x)!(N-M-(K-x))!}}{\frac{N!}{K!(N-K)!}}\\
=&\frac{K!(N-K)!}{N!}\frac{M!}{x!(M-x)!}\frac{(N-M)!}{(K-x)!(N-M-(K-x))!}\\
=&\frac{K!}{x!(K-x)!} \times \frac{1}{N(N-1)\cdots(N-(K-1))}\\
&\times \frac{M(M-1)\cdots(M-(x-1))}{1}\\
&\times \frac{(N-M)\cdots (N-M-(K-x-1))}{1}\\
=&\left(\begin{array}{c} K \\ x \\ \end{array} \right)\frac{\overbrace{M(M-1)\cdots(M-(x-1))}^{x\text{個}}\times\overbrace{(N-M)\cdots (N-M-(K-x-1))}^{K-x\text{個}}}{\underbrace{N(N-1)\cdots(N-(K-1))}_{K\text{個}}}\\
=&\left(\begin{array}{c} K \\ x \\ \end{array} \right)\frac{\frac{M}{N}(\frac{M}{N}-\frac{1}{N})\cdots(\frac{M}{N}-\frac{x-1}{N})\times(1-\frac{M}{N})\cdots (1-\frac{M}{N}-\frac{K-x-1}{N})}{1(1-\frac{1}{N})\cdots(1-\frac{K-1}{N})}\\
\rightarrow &~\left(\begin{array}{c} K \\ x \\ \end{array} \right)p^x(1-p)^{K-x}
\end{align}
\]
(証明終)

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