Let (X, Y, Z) be independent standard normal random variables. Let (, , R) be the corresponding

Let (X, Y, Z) be independent standard normal random variables. Let (Φ, Θ, R) be the corresponding spherical coordinates. The correspondence between rectangular and spherical coordinates is given by

 x = rsinφcosθ, y = rsinφsinθ, z = rcosφ, 

for 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2π, and r > 0. Find the joint density of (Φ, Θ, R) and the marginal density of R.