The meaning of E[X|Y] = E[X]

Question

In class today it was mentioned that when E[X|Y] = E[X] , the two random variables X and Y are uncorrelated, i.e. cov(X, Y) = 0.

I'm curious as to why that's the case.

Why are the two variables X and Y uncorrelated when E[X|Y] = E[X]? The professor just mentioned it as passing as if it was a natural thing, but I can't really understand how this property is proved.

Could anyone provide an explanation regarding why "E[X|Y] = E[X]" means that "covariance of X and Y = 0 ( cov(X,Y)=0 )"? Thanks.

Answer 1

By the tower rule ($E[E[Z |X]]=E[Z]$) we can write

$$ E[ E[X Y | Y ] ] = E[X Y]$$

But $E[XY |Y] = Y E[X|Y]$ and $E[X|Y]=E[X]$ by hypothesis. Then the result follows.

Notice that the reverse implication is not true.

Answer 2

$$E[XY]=E[E[XY|Y]]=E[Y\underbrace{E[X|Y]}_{E[X]}]=E[X]E[Y]$$