Deformed Normal Distributions

Abstract

It is well known that the probability distribution of the product of two normal variables itself approaches a normal distribution if one of the factor variables has a low coefficient of variation. A related, but apparently somewhat neglected problem, is to find the conditional distribution of a factor variable when the product of two independent normal variables is known. In the present paper we indicate why this conditional distribution also tends towards a normal distribution under similar conditions, and demonstrate numerically that this is indeed the case. Power series expansions for the mean and variance of the conditional distribution are also presented, which hold some problems of convergence but nevertheless provide good approximations when one coefficient of variation is low. Finally, a simplified version is presented of an actual application in object tracking, yielding an approximation for the probability distribution of the distance out to an object when the relative azimuth is known. The power series expansions shown in this paper are most conveniently developed in a more abstract setting, yielding results about the more general notion of a deformed normal distribution.