Let (X,Y) be uniformly distributed in the circle of radius 1 centered at the origin. Its j...

Let (X,Y) be uniformly distributed in the circle of radius 1 centered at the origin. Its joint density is thus

Let R = (X2  Y2)1/2 and = tan-1 (Y / X) denote the polar coordinates of (X, Y). Show that R and are independent, with R2 being uniform on (0, 1) and  being uniform on (0,2).