Are squares of independent random variables independent?

Question

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.

Answer 1

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 $0$.