Some Problems Connected with the Distributions of Products and Ratios

Abstract

In this paper, the distributions of products and ratios involving real scalar positive variables, symmetric products and symmetric ratios of real positive definite matrices and symmetric product and symmetric ratios of Hermitian positive definite matrices in the complex domain are considered. Real scalar variable case is taken as a starting point. Here, it is pointed out how distributions of products and ratios are connected to Bessel integrals, reaction-rate probability integrals in nuclear reaction-rate theory, inverse Gaussian density, Kr¨atzel integral and transform, pathway transform, fractional integrals of the first and second kind etc. Only one
illustrative example each is given. Then, symmetric product and symmetric ratios of positive definite matrices in the real and complex domain are considered. Here also one illustrative example each is given. Some current research materials are also
mentioned in the introduction.