The question in the book was:
Give an example of a pair of dependent and joinlty continuous random variables $X,Y$ for which $E(XY)=E(X)E(Y)$.
Example given: Let $X$ and $Z$ be independent. $X$ has normal distribution with mean $0$ and variance $1$. $Z$ takes the value $1$ or $-1$ with probability $1/2$. Now define $Y=XZ$.
My question:
So we know that $E(Z)=0$, so $E(XZ)=0$. But how can I see that $E(XY)$ (so in this case $E(X(XZ))$ is $0$? $X$ and $XZ$ are dependent so I cant split them into $E(X)E(XZ)$ and say that that implies that $E(X(XZ))$ is $0$.
Thanks in advance
$\mathsf E(X^2Z)=\mathsf E(X^2)~\mathsf E(Z)$ due to the independency of $X^2$ and $Z$.