Independence and Conditional Expectations

Question

I have a simple question.

It is known that, if X and Y are independent, then

$\mathbb{E}[X | Y] = E[X]$

where X and Y are just random variables.

However, is the converse true? i.e. Given that $\mathbb{E}[X | Y] = E[X]$, can I assume they are independent?

Thanks

Answer 1

Assume $\mathsf E(X\mid Y)=\mathsf E(X)$ , then $\mathsf {Var}(X,Y)~{=\mathsf E(\mathsf E(XY\mid Y))-\mathsf E(X)\mathsf E(Y) = 0}$.   Therefore $X$ and $Y$ are not linearly correlated.

Independent random variables are not linearly correlated, however the converse is not necessarily so.   Linearly uncorrelated random variables need not be independent.   Counterexamples can easily be found.   Here is one.

Assume $X\sim\mathcal U\{-2,-1,1,2\}$ and $Y=X^2$ surely.   $X$ and $Y$ are clearly dependent, but you will find that $\mathsf E(X\mid Y)=\mathsf E(X)$.

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