不偏分散の期待値

不偏分散の期待値

$$\begin{array}{rclcl} \hat{\sigma}^2 &=& \href{https://shikitenkai.blogspot.com/2019/07/specimen-random-variable.html}{\displaystyle \frac{1}{n-1}\sum_{k=1}^{n}\left( \displaystyle X_k - \overline{X} \displaystyle \right)^2}\,\dotso\,不偏分散(unbiased \, variance)\\ E\left[\hat{\sigma}^2\right] &=&E\left[ \displaystyle\frac{1}{n-1}\sum_{k=1}^{n} \left(X_k -\overline{X}\right)^2 \right]\\ &=&\displaystyle\frac{1}{n-1}E\left[\sum_{k=1}^{n} \left(X_k -\overline{X}\right)^2 \right]\\ &=&\displaystyle\frac{1}{n-1}\left(n-1\right)\sigma^2 \,\dotso\,\displaystyle \href{https://shikitenkai.blogspot.com/2019/07/overlinex2.html}{E\left[\sum_{k=1}^{n} \left(X_k -\overline{X}\right)^2 \right]=\left(n-1\right)\sigma^2}\\ &=&\displaystyle\sigma^2\\ \end{array}$$