Question

“Y的无条件方差等于Y的条件方差的期望与Y的条件期望的方差之和”这一性质的推导过程

Answer 1

DY=EY^2-(EY)^2

DE[Y|F]=E(E[Y|F])^2-(EY)^2

DY-DE[Y|F]=EY^2-E(E[Y|F])^2

条件方差

E[Y-E[Y|F]]^2

=E[Y^2-2YE[Y|F]+(E[Y|F])^2]

=EY^2-2E[YE[Y|F]+(E[Y|F])^2

=EY^2-2EE[[YE[Y|F]|F]+(E[Y|F])^2

=EY^2-2(E[Y|F])^2+(E[Y|F])^2

=EY^2-(E[Y|F])^2

所以DY-DE[Y|F]=E(Y-E[Y|F])^2

DY=E(Y-E[Y|F])^2+DE[Y|F]