How to prove that $\operatorname{var}(X) = \operatorname{var}(\mathbb E(X\mid Y)) + \mathbb E(\operatorname{var}(X\mid Y))$?

Question

Hey can someone tell me how to start, I have no idea, I need to prove the following:

$$ \operatorname{var}(X) = \operatorname{var}(\mathbb E(X\mid Y)) + \mathbb E(\operatorname{var}(X\mid Y)) $$

Answer 1

Assuming you know the double expectation formula, see http://en.wikipedia.org/wiki/Law_of_total_variance#Proof . If you don't know the double expectation formula, see http://en.wikipedia.org/wiki/Law_of_total_expectation#Proof_in_the_discrete_case (and switch it to continuous using integrals if needed).