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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?

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    $\begingroup$ 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? $\endgroup$
    – Dave
    May 30, 2020 at 19:04
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    $\begingroup$ 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. $\endgroup$
    – Regio
    Jun 1, 2020 at 15:16

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