Question

Conditional expectation:
Proving \(E[E(X|Y)|Y]=E(X|Y)\)

Answer 1

I attempted the following. Is it correct?\[E[E(X|Y)|Y=y]=\int_{-\infty}^{\infty}E(X|Y=y)f_{X|Y}(x|y)~dx\\~~~~~~~~~~~~~~~~~~~~~~~~~~~=E(X|Y=y)\int_{-\infty}^{\infty}f_{X|Y}(x|y)~dx\\~~~~~~~~~~~~~~~~~~~~~~~~~~~=E(X|Y=y)\] Hence, \(E[E(X|Y)|Y]=E(X|Y)\)

Answer 2

looks fine

You could just as easily show that \(E[G(Y)|Y]=G(Y)\)

which the above is an example of.

Answer 3

Oh that makes sense. Thanks :)