Let X and Y have a bivariate standard normal distribution with correlation ρ. Find P(X > 0, Y > 0) by the following steps.
(b) Think geometrically. Express the event as a region in the plane.
(c) See the last Exercise 10.7. Use rotational symmetry and conclude that
Exercise 10.7
Let X and Y have a bivariate standard normal distribution with correlation ρ = 0. That is, X and Y are independent. Let (x, y) be a point in the plane. The rotation of (x, y) about the origin by angle θ gives the point
Show that the joint density of X and Y has rotational symmetry about the origin. That is, show that f(x, y) = f(u, v).