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.
Are squares of independent random variables independent?
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0@Dilip: You could post that as an answer so the question doesn't remain unanswered. – 2012-10-24
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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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0@Silverfish You $a$re $a$bsolutely correct. Thanks for proof-reading carefully. I have corrected the typos. – 2013-11-06