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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9Correlation 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 Aug 28 '17 at 16:12
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2The question is more interesting without the zero mean assumptions – wolfies Aug 28 '17 at 16:31
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1Thank 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. – Roberto Aug 29 '17 at 09:40
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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
Edit Answer to question in comment:
does $X,Y,Z$ being independent implies that $X^2, Y, Z$ are too? If so, how?
Yes, functions of independent variables are independent, see Functions of Independent Random Variables, Mutual independence of functions of random variables
kjetil b halvorsen
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1Sorry I know this is a really stupid question, but how do you go from $E(XY XZ)$ to $E(X^2)E(Y)E(Z)$? To do so, it seems that $E(XY XZ)=E(XX YZ)=E(X^2 YZ)=E(X^2)E(Y)E(Z)$. I know that $X,Y,Z$ are independent, but it seems we need to know that $X^2, Y, Z$ are independent in order to get $E(X^2 YZ)=E(X^2)E(Y)E(Z)$; does $X,Y,Z$ being independent implies that $X^2, Y, Z$ are too? If so, how? – Constantly confused Sep 15 '23 at 23:01
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