Two random vars, finite mean and variance, represent Var(Y) with conditional expectations

Question

This was asked as an self-assessment question, that I was quite embarrased by, as I had no idea how to start it...

Consider two random variables X and Y that are allowed to be correlated and whose first and second moments are assumed to be finite. Show that:

$$Var(Y) = \mathbf{E}_{x}[Var(Y|X)] + Var(\mathbf{E}_{y|x}[Y|X])$$

where $\mathbf{E}_{X}$ and $\mathbf{E}_{y|x}$ denote expectations with respect to the marginal distribution of X and the conditional distribution of Y , given X, respectively.

Answer 1

$Var(Y)=EY^2-(EY)^2=E(E(Y^2|X))-(E(E(Y|X)))^2=E(Var(Y|X))+E((E(Y|X))^2)-(E(E(Y|X)))^2=E(Var(Y|X))+Var(E(Y|X))$