海风游戏狂欢季-独家活动首发平台

在有些时候，直接计算随机变量的方差非常麻烦，此时可以用方差分解公式，将方差分解为条件期望的方差加条件方差的期望：

\[\text{Var}(X)=\text{Var}[\text{E}(X|Y)]+\text{E}[\text{Var}(X|Y)]

\]

证明非常简单，注意到

\[\begin{aligned}

\text{Var}[\text{E}(X|Y)] =& \text{E}\left\{\left[\text{E}(X|Y)\right]^2\right\} - \left\{\text{E}\left[\text{E}(X|Y)\right]\right\}^2\\

=& \text{E}\left\{\left[\text{E}(X|Y)\right]^2\right\} - \left[\text{E}(X)\right]^2

\end{aligned}

\]

和

\[\begin{aligned}

\text{E}[\text{Var}(X|Y)] =& \text{E}\left\{\text{E}(X^2|Y) - [\text{E}(X|Y)]^2\right\}\\

=& \text{E}(X^2) - \text{E}\left\{\left[\text{E}(X|Y)\right]^2\right\}

\end{aligned}

\]

将上面两式相加，即得证。