二項分布の分散

二項分布 \(B(n, p)\)

$$B(n, p) = f_X(x) = \begin{cases} \displaystyle _nC_x\,p^x(1-p)^{n-x} & \quad x \in \left\{0,1,2, \dotsc ,n\right\}\\ \displaystyle　0 & \quad x \notin \left\{0,1,2, \dotsc ,n\right\} \end{cases} $$

二項分布の分散

$$\begin{array}{rcl} E[X(X-1)]&=&\displaystyle \sum_{x=0}^{n}\left(x\left(x-1\right)\right)\left( _nC_x\,p^x(1-p)^{n-x} \right)\\ &=& \displaystyle \sum_{x=0}^{n}\left(x\left(x-1\right)\right)\left(\left( \frac{n!}{x!(n-x)!}\right)p^x(1-p)^{n-x}\right)\\ &=& \displaystyle \sum_{x=0}^{n} \frac{n!}{(x-2)!(n-x)!}p^x(1-p)^{n-x}\dotso \frac{x^2}{x!}=\frac{1}{(x-2)!}\\ &=& \displaystyle \sum_{x=0}^{n} \frac{n(n-1)(n-2)!}{(x-1)!(n-x)!}p^x(1-p)^{n-x}\dotso n!=n(n-1)(n-2)!\\ &=& \displaystyle \sum_{x=0}^{n} \frac{n(n-1)(n-2)!}{(x-1)!(n-x)!}p^2p^{x-2}(1-p)^{n-x}\dotso p^x=p^2p^{x-2}\\ &=& \displaystyle \sum_{x=0}^{n} \frac{n(n-1)(n-2)!}{(x-2)!(n-x))!}p^2 p^{x-2}(1-p)^{n-x}\\ &=& \displaystyle n(n-1)p^2\sum_{x=0}^{n} \frac{(n-2)!}{(x-2)!(n-2+2-x))!} p^{x-2}(1-p)^{n-2+2+x}\\ &=& \displaystyle n(n-1)p^2\sum_{x=0}^{n} \frac{(n-2)!}{(x-2)!(n-2-(x-2))!} p^{x-2}(1-p)^{n-2-(x-2)}\dotso 2-x=-(x-2)\\ &=& \displaystyle n(n-1)p^2 \sum_{x'=0}^{n-2} \frac{(n-2)!}{x'!(n-2-x')!}p^{x'}(1-p)^{n-2-x'}\dotso x'=x-2\\ &=& \displaystyle n(n-1)p^2\,1 \dotso B(n-2, p)の総和は1に等しい．\\ &=& \displaystyle n(n-1)p^2\\ V[X]&=&\displaystyle E[X^2]-E[X]^2\\ &=&\displaystyle E[X^2]-E[X]+E[X]-E[X]^2\\ &=&\displaystyle E[X^2-X]+E[X]-E[X]^2\dotso E[X] \pm E[Y]=E[X \pm Y]\\ &=&\displaystyle E[X(X-1)]+E[X]-E[X]^2\\ &=&\displaystyle n(n-1)p^2+np-(np)^2\\ &=&\displaystyle n^2p^2-np^2+np-n^2p^2\\ &=&\displaystyle -np^2+np\\ &=&\displaystyle np(-p+1)\\ &=&\displaystyle np(1-p)\\ &=& \sigma^2 \dotso 母集団の分散\\ \end{array}$$