Probability | Part IA, 2008

Let XX and YY have the bivariate normal density function

f(x,y)=12π1ρ2exp{12(1ρ2)(x22ρxy+y2)},x,yR,f(x, y)=\frac{1}{2 \pi \sqrt{1-\rho^{2}}} \exp \left\{-\frac{1}{2\left(1-\rho^{2}\right)}\left(x^{2}-2 \rho x y+y^{2}\right)\right\}, \quad x, y \in \mathbb{R},

for fixed ρ(1,1)\rho \in(-1,1). Let Z=(YρX)/1ρ2Z=(Y-\rho X) / \sqrt{1-\rho^{2}}. Show that XX and ZZ are independent N(0,1)N(0,1) variables. Hence, or otherwise, determine

P(X>0,Y>0).\mathbb{P}(X>0, Y>0) .

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