Thread: E(X^2) exists =>E(E(X|Y) ^2) exists?

True/False?

Let X,Y by random variables. If X² has an expectation (not necessarily finite), then (E(X|Y))² has an expectation.
In other words, if E(X² ) exists, does E(E(X|Y) E(X|Y)) exist?

If X itself hasn't an expectation, the condition expectation is not well-defined.

But X does have an expectation in this case, since X^2 does.

I meant a finite expectation (of the absolute value for ). If the expectation of is finite the result holds using Jensen's inequality. So we have to deal with the case .

Could you please indicate how Jensen's inequality can be used to prove the claim under the assumption that E(X^2) is finite?

By Jensen's inequality so .