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I have three independent random variables X, Y and Z, uncorrelated between each other. Y and Z have zero mean and unit variance, X has zero mean and given variance. Do you know how to compute the correlation between the products XY and XZ? And whether it can be zero under any specific condition?

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    $\begingroup$ Correlation is proportional to the covariance $$\operatorname{Cov}(XY,XZ)=E(XYXZ) - E(XY)E(XZ) = E(X^2)E(Y)E(Z) - E(X)E(Y)E(X)E(Z).$$ (Your independence assumptions justify the second equality.) Can you go on from there? $\endgroup$ – whuber Aug 28 '17 at 16:12
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    $\begingroup$ The question is more interesting without the zero mean assumptions $\endgroup$ – wolfies Aug 28 '17 at 16:31
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    $\begingroup$ Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. $\endgroup$ – gung - Reinstate Monica Aug 28 '17 at 17:20
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    $\begingroup$ Thank you whuber, I thus understand that the correlation is null. It was not a question from a course or textbook, I'm just developing a mixed-effect model for the data I'm working on and I am not good in statistics as you can see. I didn't know about [ self-study ] anyway, thank you for that. $\endgroup$ – Roberto Aug 29 '17 at 9:40
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Partially answered in comments:

Correlation is proportional to the covariance $$\operatorname{Cov}(XY,XZ)=E(XYXZ) - E(XY)E(XZ) = E(X^2)E(Y)E(Z) - E(X)E(Y)E(X)E(Z)$$ (Your independence assumptions justify the second equality.) Can you go on from there? – whuber

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