平均、分散、共分散の計算

$X_{1} = a_{11} Y_{1} + a_{12} Y_{2} + μ_{1}$

$X_{2} = a_{21} Y_{1} + a_{22} Y_{2} + μ_{2}$

ただし，確率変数 $Y_{1}$ ， $Y_{2}$ は，互いに独立で

$Y_{1} \sim N (0, 1)$ ， $E (Y_{1}) = 0$ ， $Var (Y_{1}) = 1$

$Y_{2} \sim N (0, 1)$ ， $E (Y_{2}) = 0$ ， $Var (Y_{2}) = 1$

とする．

$X_{1}$ ， $X_{2}$ の平均 $E (X_{1})$ ， $E (X_{2})$ ，分散 $Var (X_{1})$ ， $Var (X_{2})$ ，共分散 $Cov (X_{1}, X_{2})$ を求める．

$E (X_{1}) = E (a_{11} Y_{1} + a_{12} Y_{2} + μ_{1})$

$= a_{11} E (Y_{1}) + a_{12} E (Y_{2}) + μ_{1}$

$= μ_{1}$

$E (X_{2}) = E (a_{21} Y_{1} + a_{22} Y_{2} + μ_{2})$

$= a_{21} E (Y_{1}) + a_{22} E (Y_{2}) + μ_{2}$

$= μ_{2}$

$Var (X_{1}) = Var (a_{11} Y_{1} + a_{12} Y_{2} + μ_{1})$

$= a_{11}^{2} Var (Y_{1}) + a_{12}^{2} Var (Y_{2})$

$= a_{11}^{2} + a_{12}^{2}$

$Var (X_{2}) = Var (a_{21} Y_{1} + a_{22} Y_{2} + μ_{2})$

$= a_{21}^{2} Var (Y_{1}) + a_{22}^{2} Var (Y_{2})$

$= a_{21}^{2} + a_{22}^{2}$

$Cov (X_{1}, X_{2}) = E ((X_{1} - E (X_{1})) (X_{2} - E (X_{2})))$ （共分散を参照）

$= E ((a_{11} Y_{1} + a_{12} Y_{2} + μ_{1} - μ_{1}) (a_{21} Y_{1} + a_{22} Y_{2} + μ_{2} - μ_{2}))$

$= E ((a_{11} Y_{1} + a_{12} Y_{2}) (a_{21} Y_{1} + a_{22} Y_{2}))$

$= E (a_{11} a_{21} Y_{1}^{2} + (a_{11} a_{22} + a_{12} a_{21}) Y_{1} Y_{2} + a_{12} a_{22} Y_{2}^{2})$

$= a_{11} a_{21} E (Y_{1}^{2}) + (a_{11} a_{22} + a_{12} a_{21}) E (Y_{1} Y_{2}) + a_{12} a_{22} E (Y_{2}^{2})$

$= a_{11} a_{21} + a_{12} a_{22}$

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最終更新日： 2026年7月10日