二項分布の分散

二項分布 $B(n,p)$

$$B(n,p)=f_X(x)=\{\begin{array}{ll}{\displaystyle {}_n{C}_x{p}^x(1-p{)}^{n-x}}& x\in \{0,1,2,\dots ,n\}\\ {\displaystyle 0}& x\notin \{0,1,2,\dots ,n\}\end{array}$$

二項分布の分散

$$\begin{array}{rcl}E[X(X-1)]& =& {\displaystyle \sum _{x=0}^n\left(x(x-1)\right)({}_n{C}_x{p}^x(1-p{)}^{n-x})}\\ & =& {\displaystyle \sum _{x=0}^n\left(x(x-1)\right)(\left(\frac{n!}{x!(n-x)!}\right)p^x(1-p{)}^{n-x})}\\ & =& {\displaystyle \sum _{x=0}^n\frac{n!}{(x-2)!(n-x)!}{p}^x(1-p{)}^{n-x}\dots \frac{x^2}{x!}=\frac{1}{(x-2)!}}\\ & =& {\displaystyle \sum _{x=0}^n\frac{n(n-1)(n-2)!}{(x-1)!(n-x)!}{p}^x(1-p{)}^{n-x}\dots n!=n(n-1)(n-2)!}\\ & =& {\displaystyle \sum _{x=0}^n\frac{n(n-1)(n-2)!}{(x-1)!(n-x)!}{p}^2{p}^{x-2}(1-p{)}^{n-x}\dots p^x=p^2{p}^{x-2}}\\ & =& {\displaystyle \sum _{x=0}^n\frac{n(n-1)(n-2)!}{(x-2)!(n-x))!}{p}^2{p}^{x-2}(1-p{)}^{n-x}}\\ & =& {\displaystyle n(n-1)p^2\sum _{x=0}^n\frac{(n-2)!}{(x-2)!(n-2+2-x))!}{p}^{x-2}(1-p{)}^{n-2+2+x}}\\ & =& {\displaystyle n(n-1)p^2\sum _{x=0}^n\frac{(n-2)!}{(x-2)!(n-2-(x-2))!}{p}^{x-2}(1-p{)}^{n-2-(x-2)}\dots 2-x=-(x-2)}\\ & =& {\displaystyle n(n-1)p^2\sum _{x^{\prime }=0}^{n-2}\frac{(n-2)!}{x^{\prime }!(n-2-x^{\prime })!}{p}^{x^{\prime }}(1-p{)}^{n-2-x^{\prime }}\dots x^{\prime }=x-2}\\ & =& {\displaystyle n(n-1)p^{2}1\dots B(n-2,p)\mathrm{の}\mathrm{総}\mathrm{和}\mathrm{は}1\mathrm{に}\mathrm{等}\mathrm{し}\mathrm{い}．}\\ & =& {\displaystyle n(n-1)p^2}\\ V[X]& =& {\displaystyle E[X^2]-E[X{]}^2}\\ & =& {\displaystyle E[X^2]-E[X]+E[X]-E[X{]}^2}\\ & =& {\displaystyle E[X^2-X]+E[X]-E[X{]}^2\dots E[X]\pm E[Y]=E[X\pm Y]}\\ & =& {\displaystyle E[X(X-1)]+E[X]-E[X{]}^2}\\ & =& {\displaystyle n(n-1)p^2+np-(np{)}^2}\\ & =& {\displaystyle n^2{p}^2-n{p}^2+np-n^2{p}^2}\\ & =& {\displaystyle -n{p}^2+np}\\ & =& {\displaystyle np(-p+1)}\\ & =& {\displaystyle np(1-p)}\\ & =& {\sigma }^2\dots \mathrm{母}\mathrm{集}\mathrm{団}\mathrm{の}\mathrm{分}\mathrm{散}\end{array}$$