1
$\begingroup$

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?

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.