Originally Posted by
Moo Hello,
Sorry I won't write it in Latex, I hope it will be enough.
The mean of a variable is not a variable anymore. It is a constant.
True, but the conditional mean given a random variable is a random variable (there probably are another few chapters before you get to that cool notion ).
To prove the equality, one should make the definitions of the right-hand side explicit: since $\displaystyle {\rm Var}(Z)=E[Z^2]-E[Z]^2$,
$\displaystyle E[{\rm Var}(X|Y)]+{\rm Var}(E[X|Y])=E[E[X^2|Y]-E[X|Y]^2]+E[E[X|Y]^2]-E[E[X|Y]]^2.$
Now, you should notice that two terms simplify, and if you remember that $\displaystyle E[E[Z|Y]]=E[Z]$ for any integrable $\displaystyle Z$, you'll notice that you're done.