var(X)=var[E(X|Y)]+E[var(X|Y)]怎么证明

题目

var(X)=var[E(X|Y)]+E[var(X|Y)]怎么证明

答案

由于 x09E(X) = ∫E(X|Y=y)*p(y) dy = E[E(X|Y)]
因此有：x09x09E[E(X|Y)] = E(X); x09x09①
同理：x09x09E[E(X²|Y)] = E(X²)x09x09②
下面计算
Var[E(X|Y)] + E[var(X|Y)]
= E{E(X|Y)-E[E(X|Y)]}² + E{E(X²|Y)-[E(X|Y)]²}
= E[E(X|Y)]²-{E[E(X|Y)]}² + E{E(X²|Y)-[E(X|Y)]²}
= E[E(X|Y)]²-{E[E(X|Y)]}² + E[E(X²|Y)]-E[E(X|Y)]²
= E[E(X²|Y)]-{E[E(X|Y)]}² x09x09(第一项和第四项相消,剩2,3项)
= E(X²)- E(X)²x09x09x09x09x09(由①②)
= Var(X)