Double Expectation: Proving E(aX|Y) = aE(X|Y)

[oyth94]

For two random variables X and Y , we can define E(X|Y) to be the function of Y that satisfies
E(Xg(Y)) = E(E(X|Y)g(Y))
for any function g. Using this definition show that E(aX|Y ) = aE(X|Y ).

so according to the definition do i start with this?:

E(aX|Y) = E(aE(X|Y)|Y)
if so how do i continue from there?

 

[Valkarie]

Yes, you start with this. Now use the definition to show that E(aE(X|Y)|Y) = aE(X|Y):Let g(Y) = aE(X|Y). ThenE(aX|Y) = E(g(Y)) = E(aE(X|Y)|Y) = aE(E(X|Y)|Y) = aE(X|Y)