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

$$\begin{array}{rcl}\mathrm{標}\mathrm{本}\mathrm{確}\mathrm{率}\mathrm{変}\mathrm{数}& :& X_k(k=1,2,\dots ,n)\\ \mathrm{標}\mathrm{本}\mathrm{平}\mathrm{均}(samplemean)& :& \bar{X}={\displaystyle \frac{1}{n}\sum _{k=1}^n{X}_k}\end{array}$$ $$\begin{array}{rcl}\mu & =& E[X]\\ E[X_k]& =& E[X]& =& \mu \\ E[cX]& =& cE[X]\end{array}$$ $$\begin{array}{rcl}E[\bar{X}]& =& E[{\displaystyle \frac{1}{n}\sum _{k=1}^n{X}_k]}\\ & =& {\displaystyle \frac{1}{n}E[\sum _{k=1}^n{X}_k]}\\ & =& {\displaystyle \frac{1}{n}\sum _{k=1}^{n}E[X_k]}\\ & =& {\displaystyle \frac{1}{n}\sum _{k=1}^n\mu }\\ & =& {\displaystyle \frac{1}{n}n\mu }\\ & =& {\displaystyle \mu }\end{array}$$