Normal functions of normal random variables (Q1096956)

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 \(\sigma\) and \(\tau\), respectively, then \[ Z=(\sigma^{-1}+\tau^{-1})XY/(X^ 2+Y^ 2)^{1/2}\quad and \] \[ W=sign(X)\cdot (\sigma^{-1}X^ 2-\tau^{-1}Y^ 2)/(X^ 2+Y^ 2)^{1/2} \] 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.