Does E(X|Y)=E(X) implies independence between X and Y?

Question

if anyone could help me with this doubt, it would be great! I know that independence between X and Y implies E(X|Y)=E(X) and E(Y|X)=E(Y), but i really don't know if the opposite is valid.

Thank very much for your attention! Regards

Answer 1

No, for independence you need that fact for every function $g(X)$. If it only holds for the expected value then you usually say that variables are "mean-independent".

Answer 2

No. Let $Y$ be a coin flip ($1$ wp $1/2$ and $-1$ wp $1/2$). Let $X=YU$ where $U$ is uniform $[-1,1]$ and independent of $Y$.