## kirbykirby Conditional expectation: Proving $$E[E(X|Y)|Y]=E(X|Y)$$ 8 months ago 8 months ago
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, $$E[E(X|Y)|Y]=E(X|Y)$$