Let x,y be continous random variables.
Prove that E(E(y|x)) = E(y)
I am sure this is a very basic question but still if someone can provide/guide towards a rigourous proof for the same?
Thanks
It's true in any case, but the continuous case is easy...
$\displaystyle E(Y|X)=\int yf(y|X)dy$ which is a function of X
next integrate wrt x and multiply the densities...
$\displaystyle E(E(Y|X))=\int\left(\int yf(y|x)dy\right)f_X(x)dx$
Fubini... $\displaystyle =\int\int yf(y|x)dyf_X(x)dx=\int\int yf(x,y)dxdy=E(Y)$