TITLE: Prove $Cov(X,Y)=0$ if $E(X\mid Y=y)$ is a constant
QUESTION [1 upvotes]: Let $X$ and $Y$ be two random variables. Suppose $E(X\mid Y=y)$ is a finite constant $c$. Show that $Cov(X,Y)=0$.

What I know:
$Cov(X,Y)=E(XY)-E(X)E(Y)$.
$$E(X|Y=y) = \sum_{x} xp_{X|Y}(x/y)$$
I am give that the lower term is a constant but nothing more.
I know that
$$E(X|Y=y) = E(X)$$ when both X and Y are independent but I think that is useless here becasue we are not given that $ X$ and $Y$ are independent.

REPLY [1 votes]: $\mathbb{E}[X]=\mathbb{E}[\mathbb{E}[X|Y]]=C$
$\mathbb{E}[XY]=\mathbb{E}[\mathbb{E}[XY|Y]]=\mathbb{E}[Y\mathbb{E}[X|Y]]=C\mathbb{E}[Y]=\mathbb{E}[X]\mathbb{E}[Y]$
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