標本平均の期待値と分散の導出

（i）期待値の性質 $E\left(\sum _{i=1}^{n}a{X}_i\right)=a\sum _{i=1}^{n}E\left(X_i\right)$ より、 $$\begin{array}{rl}E\left(\overline{X}\right)& =E\left[\frac{1}{n}(X_1+X_2+\cdots +X_n)\right]\\ & =\frac{1}{n}E(X_1+X_2+\cdots +X_n)\\ & =\frac{1}{n}\{E\left(X_1\right)+E\left(X_2\right)+\cdots +E\left(X_n\right)\}\\ & =\frac{1}{n}\cdot n\mu \\ & =\mu \end{array}$$ $◼$

（ii）分散の性質 $V\left(\sum _{i=1}^{n}a{X}_i\right)=a^2\sum _{i=1}^{n}V\left(X_i\right)$ より、 $$\begin{array}{rl}V\left(\overline{X}\right)& =V\left[\frac{1}{n}(X_1+X_2+\cdots +X_n)\right]\\ & =\frac{1}{n^2}V(X_1+X_2+\cdots +X_n)\\ & =\frac{1}{n^2}\{V\left(X_1\right)+V\left(X_2\right)+\cdots +V\left(X_n\right)\}\\ & =\frac{1}{n^2}\cdot n{\sigma }^2\\ & =\frac{{\sigma }^2}{n}\end{array}$$ $◼$