# Are the random variables XY and X^2 independent if X and Y are zero - mean Gaussian independent random variables?

Assume that X and Y are random variables which are normally distributed as N(0,s_X^2) and N(0,s_Y^2), respectively. Furthermore, assume that X and Y are independent. Could anyone please tell me if the variable given by the product XY and the variable X^2 are independent? Many thanks!

They're not independent. Intuitively speaking, if $$X^2=0$$ , $$XY$$ must be $$0$$, so the two have a dependence.
More formally, There are plenty of other ways to do it, but I'll focus on a simple contradiction. Let $$Z=XY,W=X^2$$, then we are asking if $$Z$$ and $$X$$ are independent. If they are, we should have $$\operatorname{var}(Z|W=w)=\operatorname{var}(Z)=\sigma_x^2\sigma_y^2$$. However, let's say $$w=0$$, then $$\operatorname{var}(Z|W=0)=0$$ because $$W=0\rightarrow X^2=0\rightarrow X=0\rightarrow XY=Z=0$$, which makes $$Z$$ deterministic, and therefore have $$0$$ variance. So, we can conclude that, in general, $$\operatorname{var}(Z|W=w)\neq\operatorname{var}(Z)$$, and the two are not independent.
• You mean "$Z$ and $W$ are independent"? Sep 27 '19 at 10:05
• Yes, $Z$ and $W$. Sep 27 '19 at 10:06