Show that E(Y | X) = E(E(Y | X, W) | X). The right hand side may sometimes also be written as E(E(Y | X, W) | X) = E(E(Y | W) | X).

Question

We know that, E(E(X|Y)) = E(v(Y)) = E(E(X|Y=y)) = E(X), where v(Y) = E(X|Y=y). Hence, E(E(X|Y)) = E(X).

Note: Let X and Y be discrete or jointly continuous random variables. The conditional expectation of X given Y, denoted by E(X|Y), is by definition the random variable v(Y) where the function v is defined by v(Y) = E(X|Y=y).