Prove that if E(X|Y) = E(X) then X and Y are uncorrelated and give an example where A and B are uncorrelated but E(A|B) isn't equal to E(A)
I know that for X and Y to be uncorrelated, correlation has to be 0. Which means Covariance has to be 0. And,
Cov(X,Y) = E(XY) - E(X) E(Y)
But this equation doesn't help get anywhere.
That equation doesn't help but if you focus on the first addend you can observe that
$$\mathbb{E}[XY]=\mathbb{E}[\mathbb{E}[XY|Y]]=\mathbb{E}[Y\underbrace{\mathbb{E}[X|Y]}_{=\mathbb{E}[X]}]=\mathbb{E}[Y]\mathbb{E}[X]$$
Thus you are all set...