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Assuming two variables $X1$ ~ $N(0,1)$, $X2$ ~ $N(0,1)$

with $Cov(X1,X2) = a$.

Is it possible to derive analytically what the covariance between $X1^2$ and $X2^2$ would be? Empirically (I tried this with large simulations), it appears that

$Cov(X1^2,X2^2) = 2*(Cov(X1,X2))^2$

I think this may however be just a coincidence because variables X1 and X2 have mean 0, and would in general love to derive this analytically.

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  • $\begingroup$ "I think this may however be just a coincidence because variables X1 and X2 have mean 0, and would in general love to derive this analytically." So why not cut to the chase and set up $X1$ ~ $N(\mu_1,\sigma_1^2)$ and $X2$ ~ $N(\mu_2,\sigma_2^2)$? $\endgroup$ – StatsStudent Mar 12 '15 at 5:27
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    $\begingroup$ it is as you suspect, there are two derivations here: math.stackexchange.com/questions/668641/… $\endgroup$ – Chris Novak Mar 12 '15 at 7:48
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Answer provided here: https://math.stackexchange.com/questions/668641/covariance-of-two-chi-square-random-variables

as indicated by Chris Novak in comments about.

Thanks, this thread did answer this question, and showed a derivation (or at least how one is obtained in Mathematica)

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