If I have a random variable X that is uncorrelated with another random variable Z, and a r.v. Y that is also uncorrelated with Z, does it follow that the product XY is uncorrelated with Z?
And what if X and Y were not only uncorrelated but independent of Z?
Thank you!
Let $X,Y = \left.\begin{cases} 1 & \text{with probability }1/2, \\ 0 & \text{with probability 1/2,} \end{cases} \right\}$ and further suppose $X,Y$ are independent. Let $$Z = (X+Y\bmod 2).$$ Then $X$ and $Z$ are independent, and $Y$ and $Z$ are independent, although $X,Y,Z$ are not independent. Now notice that $$ \operatorname{cov}(XY, Z) = \frac{-1} 8 \ne 0. $$ Whether this answers you concern about independence is perhaps not clear, since, although $Z$ is independent of $X$ and also independent of $Y$, $Z$ is not independent of the pair $(X,Y)$. If $Z$ were independent of the pair $(X,Y)$, then $Z$ would be independent of every function of $(X,Y)$, including $XY$.
However, $Z$ is uncorrelated with the pair $(X,Y)$, since correlation, unlike indpendence, is inherently pairwise. You have pairwise independence of $X,Y,Z$ without independence, but the three are uncorrelated.