# Are these independent random variables?

Let $$X, Y, Z$$ be pair-wise independent random variables. If $$A=XY$$ and $$B=XZ$$, are $$A$$ and $$B$$ independent?

My thoughts are that they are not independent. For example, if $$X$$ takes values $$-1, 1$$ with equal probability while $$Y$$ and $$Z$$ take vales $$1$$ and $$2$$ with equal probability. The probability that $$A=2$$ and $$B=-2$$ is equal to zero, while the probability that $$A=2$$ alone equals $$1/4$$ and similarly for $$B=-2$$ alone. So the joint distribution is not a product distribution.

Are there any general conditions under which $$A$$ and $$B$$ are independent?

• Is this a homework question? Please add the self-study tag and read its wiki so you know how we handle homework problems on here. What progress have you made with your homework problem so far?
– Dave
May 30, 2020 at 19:04
• Sorry, for my sloppy notation. It is not a homework problem, but it seems like it is a very simple question. I will try to clarify the notation and express my thoughts on it. to see if the question can be opened again. Sorry for the inconvenience. Jun 1, 2020 at 15:16