Let us present a two-dimensional standard normal r.vec. X in the polar coordinates as (R, ), where

Let us present a two-dimensional standard normal r.vec. X in the polar coordinates as (R, Θ), where R is the length of X and Θ is the angle between X and the first axis.

(a) Are the r.v.’s R, Θ independent? What are the marginal distributions?

(b) Would you expect the independence of R and Θ in the case of a non-standard normal distribution?