確率変数の標準化

確率変数の標準化

期待値(平均)が\(\mu\), 分散が\(\sigma^2\)の確率変数\(X\)

$$ \begin{eqnarray} \mathrm{E}\left[X\right]&=&\mu \\\mathrm{V}\left[X\right]&=&\sigma^2 \end{eqnarray} $$

確率変数の変換\(Z=\frac{X-\mu}{\sigma}\)

$$ \begin{eqnarray} \\Z&=&\frac{X-\mu}{\sigma} \end{eqnarray} $$

変換後の確率変数\(Z\)の期待値(平均)と分散

$$ \begin{eqnarray} \\\mathrm{E}\left[Z\right]&=&\mathrm{E}\left[\frac{X-\mu}{\sigma}\right] \\&=&\frac{1}{\sigma}\mathrm{E}\left[X-\mu\right] \;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-expected-value.html}{\mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]} \\&=&\frac{1}{\sigma}\left(\mathrm{E}\left[X\right]-\mu\right) \;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-expected-value.html}{\mathrm{E}\left[X\pm t\right]=\mathrm{E}\left[X\right] \pm t} \\&=&\frac{1}{\sigma}\left(\mu-\mu\right) \;\cdots\;\mathrm{E}\left[X\right]=\mu \\&=&\frac{1}{\sigma}\left(0\right) \\&=&0 \\\mathrm{V}\left[Z\right]&=&\mathrm{V}\left[\frac{X-\mu}{\sigma}\right] \\&=&\frac{1}{\sigma^2}\mathrm{V}\left[X-\mu\right] \;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-variance.html}{\mathrm{V}\left[cX\right]=c^2\mathrm{V}\left[X\right]} \\&=&\frac{1}{\sigma^2}\mathrm{V}\left[X\right] \;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-variance.html}{\mathrm{V}\left[X\pm t\right]=\mathrm{V}\left[X\right]} \\&=&\frac{1}{\sigma^2}\sigma^2 \;\cdots\;\mathrm{V}\left[X\right]=\sigma^2 \\&=&1 \end{eqnarray} $$ \(X\)の分布によらず，\(X\)の期待値(平均)と分散が\(\mu\)と\(\sigma^2\)であることから\(Z\)の期待値(平均)と分散が\(0\), \(1\)と標準化される．