$E(X|Y)=E(X)$ for NOT independent random variables

Question

is there a probability space $(\Omega, \mathcal A, P)$ with two NOT independent random variables $X,Y: \Omega \rightarrow \mathbb R$ ($X \in \mathcal L^1(P)$), such that

\begin{align} E(X|Y)=E(X). \end{align}

I was searching for a while, but I'm not able to find an example for the moment.

In addition to that, our professor mentioned, that all random variables $X,Y \in \mathcal L^2(P)$ with $E(X|Y)=E(X)$ are uncorrelated. How can I proof it?

Maybe someone can help me.

Greets

Answer 1

$X\sim\mathcal{U}[-1;1]$ and $Y=X^2$ is an example of two dependent random variables in which: $$\mathsf E(X)=0, \mathsf E(X\mid Y)=0$$

Now, if $\mathsf E(X\mid Y)=\mathsf E(X)$, then the covariance is zero.$$\begin{align}\mathsf {Cov}(X,Y) ~&=~ \mathsf E(XY)-\mathsf E(X)\mathsf E(Y) \\[1ex]&=~ \mathsf E\Big(\mathsf E\big(X\mid Y\big)~Y\Big)-\mathsf E\Big(\mathsf E\big(X\big)~Y\Big) \\[1ex]&=~ \mathsf E\Big(\mathsf E\big(X\mid Y\big)~Y-\mathsf E\big(X\big)~Y\Big) \\[1ex] &=~ 0\end{align}$$