Correlation between $X\cdot W$ and $Z$ ($Z$ and $W$ are independent)

Question

I have quite a tricky question about correlation. Suppose that there are three random variables $X$, $W$ and $Z$. $X$ and $W$ are correlated and $X$ and $Z$ are also correlated. But, $W$ and $Z$ are independent. Assume that all the three variables are mean zero and have unit variances. Then $\operatorname E[XZ]$ and $\operatorname E[XW]$ are nonzero and $\operatorname E[WZ]$ is zero. The correlation between "the product of $X$ and $W$" and "$Z$" is computed by $\operatorname E[XWZ]$. Can somebody show that $\operatorname E[XWZ]$ is equal to zero under all the assumptions? Many thanks.

Answer 1

The claim is not true. Let $W$ and $Z$ be independent with mean $0$ and variance $1$. Then $E(Z+W)^2=2$, so define $$ X:={(Z+W)^2-2\over c} $$ where $c$ is a constant chosen so that $X$ has unit variance. Then $X$, $Y$, $Z$ meet your conditions, but $$ cE(XWZ)= E\left([(Z+W)^2-2]WZ\right) = E(Z^3W)+2E(W^2Z^2)+E(W^3Z) -2E(WZ)=2, $$ assuming $W$ and $Z$ have enough finite moments to make the calculation go through.