Let X and Y have a bivariate standard normal distribution with correlation = 0. That is,

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 

(u, v) = (x cosθ − ysinθ, xsinθ + ycosθ). 

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).