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

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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.

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