Question: Independence of radial and angular parts. Let X and Y be independent normal (0, σ²) random variables. Let (R, Θ) be (X, Y) in polar coordinates, so X = R cos Θ, Y = R sin Θ.

a) Show that R and Θ are independent, and that Θ has uniform (0, 2π) distribution.

b) Let R and Θ now be arbitrary random variables such that R/σ has the Rayleigh distribution (c1), Θ has uniform (0, 2π) distribution, and R and Θ are independent. Explain why the random variables X = R cos Θ any Y = R sin Θ must be independent normal (0, σ²)

c) Find function h and k such that if U and V are independent uniform (0, 1) random variables, then X = σh(U) cos [k(V)] and Y = σh(U) sin [k(V)] are independent normal (0, σ²). [This gives a means of simulating normal random variables using a computer random number generator. Try generating a random scatter of independent bivariate normally distributed pairs if you have random numbers available, it should look like the scatter.