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If X and Y are independent random variables both with the same mean (0) and variance, how about $X^2$ and $Y^2$? I tried calculating E($X^2Y^2$)-E($X^2$)E($Y^2$) but haven't been able to get anywhere.

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They must be independent. How could you calculate $E(X^2Y^2)$? –  Berci Oct 24 '12 at 21:37
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If $X$ and $Y$ are independent, then so are $g(X)$ and $h(Y)$ for (measurable) functions $g(\cdot)$ and $h(\cdot)$. Means and variances don't come into the picture and your attempted calculation of $\text{cov}(X^2,Y^2)$ will not prove independence even though the covariance will turn out to be $0$. –  Dilip Sarwate Oct 24 '12 at 21:41
    
Thank you! That's very helpful to know. –  Jarris Oct 24 '12 at 21:56
    
@Dilip: You could post that as an answer so the question doesn't remain unanswered. –  joriki Oct 24 '12 at 22:42
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up vote 3 down vote accepted

As per joriki's suggestion, my comment (with additional information) is posted as an answer.

If $X$ and $Y$ are independent, then so are $g(X)$ and $h(Y)$ independent random variables for (measurable) functions $g(⋅)$ and $h(⋅)$. In particular, $X^2$ and $Y^2$ are independent random variables if $X$ and $Y$ are independent random variables. Means and variances don't come into the picture at all, and your attempted calculation of $\text{cov}(X^2,Y^2)$ will not prove independence even though the covariance will turn out to be $0$.

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"In particular, $X^2$ and $Y^2$ are independent random variables if $X^2$ and $Y^2$ are independent random variables." I am 99% sure that this should read "In particular, $X^2$ and $Y^2$ are independent random variables if $X$ and $Y$ are independent random variables"? –  Silverfish Nov 6 '13 at 0:20
    
@Silverfish You are absolutely correct. Thanks for proof-reading carefully. I have corrected the typos. –  Dilip Sarwate Nov 6 '13 at 2:13
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