Conditional Expectation Counterexample

Question

I am trying to think of a counterexample to the following:

If $E(Y|X) =0$, and $E(Y^2|X) = \sigma ^2$, a constant, then $X$ and $Y$ are independent.

Thanks for any help in advance.

Answer 1

Let $P(X=0)=P(X=1)=1/2$. Conditional on $X=0$ let $Y\sim N(0,1)$ and conditional on $X=1$ let $Y\sim U[-6,6]$. For example.