Uncorrelated and X given $Y = 0$

Question

Is the following true or false?

Suppose that $X$ and $Y$ are two discrete random variables defined on the same probability space. If $E[X] = E[Y] = 0$ and $E[X | Y=y] = 0$ for all $y\in Y$, then $X$ and $Y$ are uncorrelated.

How does this tell me $E[XY] = 0$?

Answer 1

$E[X|Y] = \sum_{y} E[X|Y=y] P(Y=y) = \sum_{y} 0 P(Y=y) = 0$.

Now, note $E[XY] = E[E[XY|Y]] = E[Y E[X|Y]] = E[Y (0) ] = E[0] =0$.

Answer 2

Note: it does not have to be zero.

Suppose for some constant $c$, $\forall y\in Y(\Omega): \mathsf E(X\mid Y=y)=c$.   That is $\mathsf E(X\mid Y)=c$

$\begin{align}\because \mathsf {Var}(X,Y) & = \mathsf E(X\cdot Y)-\mathsf E(X)\cdot\mathsf E(Y) \\ & = \mathsf E(\mathsf E(X\cdot Y\mid Y)) - \mathsf E(\mathsf E(X\mid Y))\cdot\mathsf E(Y) \\ & = \mathsf E(\mathsf E(X\mid Y)\cdot Y)-\mathsf E(\mathsf E(X\mid Y))\cdot\mathsf E(Y) \\ & = \mathsf E(c\cdot Y)-c\cdot \mathsf E(Y) \\ \therefore \mathsf {Var}(X,Y) & = 0 \end{align}$