# Conditional Expectation Problem with Multiple Variables

Let X be some random variable with unconditional mean 0.

Suppose I have that

$E[XY]=E[X|Y]=0$ and $E[XZ] \neq 0$

Does it follow that $E[XYZ]=0$?

Here are my thoughts:

$E[XYZ]=E[E[XYZ|Y,Z]]=E[ZE[XY|Y]|Z]=0$

where $E[XY|Y]=0$ follows from our assumptions

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In general, $E[XYZ]\ne0$. For example, let $P(X=1)=P(X=-1)=\dfrac{1}{2}$, $Y=1$ and $Z=X$.