# Conditional expectation and uncorrelated random variables

Let $$X$$ and $$Y$$ be two random variables which are uncorrelated. That is,

$$\mathsf{E}[X ~Y] = \mathsf{E}[X] ~\mathsf{E}[Y].$$

Does this mean that the conditional expectation

$$\mathsf{E}[X | Y] = \mathsf{E}[X]?$$

Note that all of these statements are true if the random variables are independent. But, it is not immediately clear to me that the second statement is true when they are uncorrelated.

Consider the following counter example: let $$Y \sim N(0, 1)$$ and $$X = Y^2$$.
Then $$X$$ and $$Y$$ are uncorrelated, as $$E[XY] = E[Y^3] = 0 = E[Y]E[Y^2]$$.
On the other hand, $$E[X|Y] = E[Y^2|Y] = Y^2 \neq E[X] = E[Y^2] = 1$$, since $$Y^2 \sim \chi_1^2$$ by construction.
For example, consider the following discrete model. $$\matrix{ x & | & y & | & \mathbb P(X=x, Y=y) \cr \hline\cr 1 & | & 0 & | & 1/4 \cr -1 & | & -1 & | & 1/2 \cr 1 & | & -2 & | & 1/4 }$$ Since $$\mathbb E[X] = 0$$ and $$\mathbb E[XY] = 0$$, $$X$$ and $$Y$$ are uncorrelated. However, $$\mathbb E[X|Y=0] = 1$$ while $$\mathbb E[X] = 0$$.