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?