If $E(Y\mid X)$ is constant then $X, Y$ are uncorrelated.

Question

Last minute studying please tell me how to:

Prove that if the expected conditional expected value of the random variable $X$ given the random variable $Y$ - denoted by $E(X\mid Y)$ - is constant then $X, Y$ are uncorrelated.

Answer 1

What you need is the double expectation formula: $$ \DeclareMathOperator{\E}{E} \E(X) = \E \E (X|Y) $$ In the double expectation, the inner expectation is a function of $Y$, that is, a random variable, and the outer expectation then is then taken over the distribution of $Y$. Assume $\E X = \E Y = 0$ (we can do that without loss of geneality, since else, just subtract first the expectation), and then $\E(X | Y)=s$, a known, constant real number. Now calculate $$ \E XY = \E (\E (XY|Y)) = \E ( Y \E(X|Y)) = \E Ys = s\E Y =0. $$

Answer 2

If ${\rm E}[X\mid Y]$ is constant, then it must necessarily be equal to its mean ${\rm E}[{\rm E}[X\mid Y]]={\rm E}[X]$. Now use that $$ {\rm E}[XY]={\rm E}[{\rm E}[XY\mid Y]] $$ to conclude that ${\rm E}[XY]={\rm E}[X]{\rm E}[Y]$ which is equivalent to saying that the covariance is zero.