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:
Var(X)=Var[E(X∣Y)]+E[Var(X∣Y)]
\text{Var}(X)=\text{Var}[\text{E}(X|Y)]+\text{E}[\text{Var}(X|Y)]
Var(X)=Var[E(X∣Y)]+E[Var(X∣Y)]
证明非常简单,注意到
Var[E(X∣Y)]=E{[E(X∣Y)]2}−{E[E(X∣Y)]}2=E{[E(X∣Y)]2}−[E(X)]2
\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}
Var[E(X∣Y)]==E{[E(X∣Y)]2}−{E[E(X∣Y)]}2E{[E(X∣Y)]2}−[E(X)]2
和
E[Var(X∣Y)]=E{E(X2∣Y)−[E(X∣Y)]2}=E(X2)−E{[E(X∣Y)]2}
\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}
E[Var(X∣Y)]==E{E(X2∣Y)−[E(X∣Y)]2}E(X2)−E{[E(X∣Y)]2}
将上面两式相加,即得证。