標本確率変数(Specimen random variable) / 標本平均の期待値

$$\begin{array}{rcl} 標本確率変数&:&X_k (k=1,2,\dotsc ,n)\\ 標本平均(sample \, mean)&:&\overline{X}=\displaystyle\frac{1}{n}\sum_{k=1}^{n} X_k\\ \end{array}$$ $$\begin{array}{rclcl} \mu&=&E[X]\\ E[X_k]&=&E[X]&=&\mu\\ 　　　　E[cX]&=&c E[X]\\ \end{array}$$ $$\begin{array}{rclcl} E[\overline{X}]&=&E[\displaystyle\frac{1}{n}\sum_{k=1}^{n} X_k]\\ &=&\displaystyle\frac{1}{n}E[\sum_{k=1}^{n} X_k]\\ &=&\displaystyle\frac{1}{n}\sum_{k=1}^{n} E[X_k]\\ &=&\displaystyle\frac{1}{n}\sum_{k=1}^{n} \mu\\ &=&\displaystyle\frac{1}{n}n\mu\\ &=&\displaystyle\mu\\ \end{array}$$