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

$$

f(x,y)= \frac{1}{\pi}

\qquad 0 \le x^2 + y^2 \le 1

$$

Let R = (X² + Y²)¹/² and θ = tan⁻¹(Y/X) denote its polar coordinates.

Show that R and θ are independent with R² being uniform on (0, 1) and θ

being uniform on (0, 2π).