25. Let (X , Y) be uniformly distributed in a circle of radius r about the origin. That is, their joint density is given by f(x,y) = A, 0 < x2 + y2 < r2 nr Let R = yJx2 + Y2 and 0 = arc tan Y/X denote their polar coordinates.

Show that R and 0 are independent with 0 being uniform on (0, 2n) and P[R < a] = a2/r2, 0 < a < r.