If $X$ and $Y$ are two uncorrelated identically distributed random variables having the Normal distribution with $0$ mean, are $X^{2}$ and $Y^{2}$ also uncorrelated?
I know this holds for independent random variables, but I could not prove it for uncorrelated ones. I also know that this holds if $(X, Y)$ jointly follow the bivariate Normal distribution, but that is not known here.
Best Answer
Hint:
Consider this frequently used example:
What is the distribution of $Y$?
What is the correlation between $X$ and $Y$?
What are the distributions of $X^2$ and $Y^2$?
What is the correlation between $X^2$ and $Y^2$?