Normal functions of normal random variables

Abstract

It is the purpose of this paper to show that, when X and Y are independent normal random variables with zero means and (possibly unequal) standard deviations σ and τ, respectively, then Z = (σ −1 + τ −1)XY/(X 2 + Y 2) 1 2 and W = sign(X)·(σ −1X 2−τ −1Y 2)/(X 2 + Y 2) 1 2 are independent normal variables, both with mean 0 and variance 1. The parts of this result which exist in the literature have proofs which are needlessly sophisticated and technical. We make use of a simple univariate transformation of a uniform variable.