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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.

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  • $\begingroup$ If $X$ and $Y$ are jointly normal then they're uncorrelated iff they're independent. Otherwise there's no reason for this to be true and you should be able to find a counterexample where $X$ and $Y$ aren't jointly normal although it may be annoying to construct. $\endgroup$ Commented Dec 23, 2020 at 8:25

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Hint:

Consider this frequently used example:

  • $X$ has a standard normal distribution with mean $0$ and variance $1$
  • $Z =+1$ or $-1$ each with probability $\frac12$, independent of $X$
  • $Y=XZ$

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 $Y^2$?

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