標本確率変数(Specimen random variable) / 標本平均の分散

$$\begin{array}{rcl}\mathrm{母}\mathrm{集}\mathrm{団}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{変}\mathrm{数}& :& X\\ \mathrm{標}\mathrm{本}\mathrm{確}\mathrm{率}\mathrm{変}\mathrm{数}& :& X_k(k=1,2,\dots ,n)\\ \mathrm{標}\mathrm{本}\mathrm{平}\mathrm{均}(samplemean)& :& \bar{X}={\displaystyle \frac{1}{n}\sum _{k=1}^n{X}_k}\end{array}$$ $$\begin{array}{rcl}{\sigma }^2& =& V[X]\dots \mathrm{母}\mathrm{集}\mathrm{団}\mathrm{の}\mathrm{分}\mathrm{散}\end{array}$$ $$\begin{array}{rcl}V[\bar{X}]& =& V[{\displaystyle \frac{1}{n}\sum _{k=1}^n{X}_k]}\\ & =& {\displaystyle {\left(\frac{1}{n}\right)}^{2}V[\sum _{k=1}^n{X}_k]\dots V[cX]=c^{2}V[X]}\\ & =& {\displaystyle \frac{1}{n^2}\sum _{k=1}^{n}V[X_k]\cdots X_k\mathrm{同}\mathrm{士}\mathrm{が}\mathrm{互}\mathrm{い}\mathrm{に}\mathrm{独}\mathrm{立}(\mathrm{Con}(X_i,X_j)=0)\mathrm{で}\mathrm{あ}\mathrm{る}\mathrm{な}\mathrm{ら}V[\sum _{k=1}^n{X}_k]=\sum _{k=1}^{n}V[X_k]}\\ & =& {\displaystyle \frac{1}{n^2}\sum _{k=1}^n{\sigma }^2\dots V[X_k]=V[X]={\sigma }^2}\\ & =& {\displaystyle \frac{1}{n^2}n{\sigma }^2}\\ & =& {\displaystyle \frac{{\sigma }^2}{n}}\end{array}$$ よって標本数$n$を増やすことで$\mathrm{V}[\bar{X}]$は$0$に近づいていく．