不偏分散の期待値

不偏分散の期待値

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

標本確率変数(Specimen random variable) / 標本平均・標本分散・不偏分散

\(n\)個の標本(試行して得られたもの)に対して求めたものが，標本平均,標本分散，不偏分散となる

$$\begin{array}{rcl} 標本確率変数&:&X_k (k=1,2,\dotsc ,n)\\ \end{array}$$ $$\begin{array}{rclcl} \overline{X}&=&\displaystyle\frac{1}{n}\sum_{k=1}^{n} X_k &\dots&標本平均(sample \, mean)\\ s^2&=&\displaystyle\frac{1}{n}\sum_{k=1}^{n} (X_k - \overline{X})^2 &\dots&標本分散(sample \, variance)\\ \hat{\sigma}^2&=&\displaystyle\frac{1}{n-1}\sum_{k=1}^{n} (X_k - \overline{X})^2 &\dots&不偏分散(unbiased \, variance)\\ \end{array}$$

全事象をもとに求めたものが母平均,母分散となる

$$\begin{array}{rclcl} \mu&=&\displaystyle\frac{1}{N}\sum_{k=1}^{N} X_i &\dots&母平均(population \, mean)\\ \sigma^2&=&\displaystyle\frac{1}{N}\sum_{i=1}^{N} (X_i - \mu)^2 &\dots&母分散(population \, variance)\\ \end{array}$$