A note on normal functions of normal random variables

AbstractIt is the purpose of this note to provide a direct proof of the fact that, when X and Y are independent, normally distributed random variables with zero means and variances σ12 and σ22 respectively, then Z = XY(X2 + Y2)12 is also normally distributed with mean zero and has standard deviation σ1σ2(σ1 + σ1). This fact has been established by appeal to the literature on stable distributions. Although this may be the “standard method of proof,” it was thought interesting to demonstrate the fact directly and at the same time, to correct a couple of typographical errors in the original article on the subject [4]. The case where σ1 ≠ σ2 was that of primary concern, since, otherwise, one could establish the fact rather easily