The Distribution of the Product of Two Central or Non-Central Chi-Square Variates

Abstract

1. Summary. The operational method of Mellin transforms is employed here to derive some interesting distribution functions. The distribution of the product of two non-central chi-square variates is obtained and some special cases exhibited. Finally, an application of the derived distributions to a problem in products of complex numbers is discussed. 2. Introduction. In many problems which arise in the physical sciences the use of complex variable notation greatly facilitates the deseription of some form of rotational motion. This is especially true in the mathematical theory of spinstabilized rockets. For statistical analysis purposes it is sometimes desirable to find an expression for the probability that the magnitude of the product of two complex numbers (whose real and imaginary parts are independent random variables) is contained in a circle of some specified radius. Such a problem stimulated the work described here. The solution of the original problem indicated the solution to a more general problem which is given here with the thought that this also may be of interest. The basic problem was to find the probability density function of a random variable w = Y1Y2 where y, is a Rayleigh variate (with scale parameter equal to one), Y2 is a non-central Rayleigh variate (with scale parameter equal to one) and y, is independent of Y2 . Since the square of a Rayleigh variate is a chi-square variate this problem was solved by obtaining the distribution of the product of a chi-square variate and a non-central chi-square variate, each with two degrees of freedom. The solution of this problem was generalized to include the distribution of the product of two non-central chi-square variates with arbitrary degrees of freedom. The distributions derived here were obtained by the use of Mellin transforms. The operational advantages of Mellin transforms in problems of this type have been described by Epstein [4], and additional information on Mellin transforms can be found in [71 and [81. 3. The distribution of the product of two non-central chi-square variates. Let us suppose that y, and Y2 are two independent random variables distributed according to the non-central X2 density function [6] with non-centrality parameters A, and A2 and degrees of freedom k, and k2 respectively. Thus the density function of yj is