Assuming a r.v. $X​$ with pdf $N(0,σ_x^2)$​ and an independent r.v $Y​$ with pdf $N(0,σ_y^2)$​

for a set of positive constants $\alpha$, $\beta$, $\gamma​$

what would be the distribution of the r.v.

$$Z= \alpha X^2 + \beta XY + \gamma Y^2​$$

The problem is more complicated with nonzero means, so I would like to tackle it step by step

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  • Notice that $X^2, XY, Y^2$ are all chi-square distributions. A sum of chi squares is chi square (might be non-central but is still chi square). – Rodrigo Zepeda Nov 25 '15 at 06:53
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    @RodrigoZepeda: Are you sure $XY$ is a chi-square distribution? – Henry Nov 25 '15 at 09:02
  • @Henry check the answer to this question: http://math.stackexchange.com/questions/101062/is-the-product-of-two-gaussian-random-variables-also-a-gaussian – Rodrigo Zepeda Nov 25 '15 at 15:22
  • @RodrigoZepeda: So the answer seems to be no, but $XY$ might be related to the difference of two scaled $\chi^2$-distributed random variables – Henry Nov 25 '15 at 15:30
  • @Henry check the second answer (the one with 20-something points) – Rodrigo Zepeda Nov 25 '15 at 15:57
  • @RodrigoZepeda: That answer is what led me to say *"the difference of two scaled $\chi^2$-distributed random variables"* – Henry Nov 25 '15 at 15:59
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    @RodrigoZepeda , the distribution of $X^2$ and $Y^2$ is chi-square, but $XY$ has a K-distribution. The difficulty in getting the total distribution is that $X^2$ and $XY$ are dependent, same for $XY$ and $Y^2$ – azaz104 Nov 25 '15 at 17:42

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