方差分解公式 已注销 2021-04-21 764 阅读1分钟 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。
