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

$$\begin{array}{rcl} 母集団の確率変数&:&X\\ 標本確率変数&:&X_k (k=1,2,\dotso ,n)\\ 標本平均(sample \, mean)&:&\overline{X}=\displaystyle\frac{1}{n}\sum_{k=1}^{n} X_k\\ \end{array}$$ $$\begin{array}{rcl} \sigma^2&=&V[X]\,\dotso\,母集団の分散\\ \end{array}$$ $$\begin{array}{rclcl} V[\overline{X}]&=&V[\displaystyle\frac{1}{n}\sum_{k=1}^{n} X_k]\\ &=&\displaystyle\left( \frac{1}{n} \right)^2 V[\sum_{k=1}^{n} X_k] \,\dotso\,\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-variance.html}{V[cX]=c^2V[X]}\\ &=&\displaystyle\frac{1}{n^2}\sum_{k=1}^{n} V[X_k] \;\cdots\;X_k同士が互いに独立\left(\mathrm{Con}\left(X_i, X_j\right)=0\right)であるならV[\sum_{k=1}^{n} X_k]=\sum_{k=1}^{n} V[X_k] \\&=&\displaystyle\frac{1}{n^2}\sum_{k=1}^{n} \sigma^2 \,\dotso\,\href{https://shikitenkai.blogspot.com/2019/06/specimen-random-variable.html}{V[X_k]=V[X]=\sigma^2}\\ &=&\displaystyle\frac{1}{n^2}n\sigma^2\\ &=&\displaystyle\frac{\sigma^2}{n}\\ \end{array}$$ よって標本数\(n\)を増やすことで\(\mathrm{V}[\overline{X}]\)は\(0\)に近づいていく．