不偏分散の期待値

不偏分散の期待値

$$\begin{array}{rcl}{\hat{\sigma }}^2& =& {\displaystyle \frac{1}{n-1}\sum _{k=1}^n{\left({\displaystyle X_k-\bar{X}{\displaystyle }}\right)}^2}\dots \mathrm{不}\mathrm{偏}\mathrm{分}\mathrm{散}(unbiasedvariance)\\ E\left[{\hat{\sigma }}^2\right]& =& E\left[{\displaystyle \frac{1}{n-1}\sum _{k=1}^n{(X_k-\bar{X})}^2}\right]\\ & =& {\displaystyle \frac{1}{n-1}E\left[\sum _{k=1}^n{(X_k-\bar{X})}^2\right]}\\ & =& {\displaystyle \frac{1}{n-1}(n-1){\sigma }^2\dots {\displaystyle E\left[\sum _{k=1}^n{(X_k-\bar{X})}^2\right]=(n-1){\sigma }^2}}\\ & =& {\displaystyle {\sigma }^2}\end{array}$$