Question

条件方差问题证明 Var(Y−X|X) = Var(Y|X)求大神给点思路

Answer 1

Var(Y-X | X) = E((Y-X)^2 | X) - (E(Y-X | X))^2

= E(Y^2 | X) - 2 E(YX | X) + E(X^2 | X) - (E(Y | X))^2 + 2 E(Y | X) E(X | X) - (E(X | X))^2

= (E(Y^2 | X) - (E(Y | X))^2) - 2 E(YX | X) + E(X^2 | X) + 2 E(Y | X) E(X | X) - (E(X | X))^2

= Var(Y | X) - 2 E(YX | X) + E(X^2 | X) + 2 E(Y | X) E(X | X) - (E(X | X))^2

因为：E(YX | X) = X E(Y | X),E(X^2 | X) = X^2,E(X | X) = X

所以：Var(Y-X | X)

= Var(Y | X) - 2 X E(Y | X) + X^2 + 2 X E(Y | X) - X^2

= Var(Y | X)