Let be uniformly distributed on and define . Show that, conditionally on
the vector is uniformly distributed on the unit disc. Let denote the point in polar coordinates and find its probability density function for . Deduce that and are independent.
Introduce the new random variables
noting that under the above conditioning, are uniformly distributed on the unit disc. The pair may be viewed as a (random) point in with polar coordinates . Express as a function of and deduce its density. Find the joint density of . Hence deduce that and are independent normal random variables with zero mean and unit variance.