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suppose $(X_1,X_2)\sim\mathcal{N}(0,0,1,1,\rho)$. find distribution $$\mathbb{Y}=(X_1^2-2\rho X_1X_2+X_2^2)$$

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up vote 2 down vote accepted

HINT: Note that $Y = (X_1 - \rho X_2)^2 + (1-\rho^2) X_2^2$. Find the joint distribution of $(X_1 - \rho X_2, \sqrt{1-\rho^2} X_2)$.

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