Normal functions of normal random variables

AbstractIt 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/(X2 + Y2)12 and W = sign(X)·(σ−1X2−τ−1Y2)/(X2 + Y2)12 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