The H-function distribution has been shown to be a powerful addition in the study of the algebra of non-negative random variables. However, most of this work has been restricted to the study of independent H-function variates. This paper introduces a bivariate probability distribution based on the H-function of two variables. The distribution is shown to be a generalization of several known bivariate distributions. Further, it is shown that products, quotients, and powers of bivariate H-function variates are H-function variates. Several examples are given. Eldred
[5], Cook [3], Mathai and Saxena [ll], and others, the H-function distribution has been shown to be a generalization of most known distributions of non-negative random variables. Through their efforts, sums, products, quotients and powers of independent random variates can be easily analyzed. Through the work of such authors as Nair [12], Fox [8], and Subrahmaniam [19], much of the work accomplished for rational combinations of independent variates can be extended to dependent variates. Fox [8] applied the use of double Mellin transforms to mixtures of two pairwise independent bivariate distributions. Subrahmaniam [19] extended this work and showed how to use the double Mellin transform for products and quotients of two dependent variables. Work on bivariate distributions using H-functions is presented by Srivastava et al. [lo]. It is the purpose of this paper to extend this work by the introduction of a bivariate probability distribution which is based on the H-function of two variables given by Goyal [lo]. This bivariate probability density function is shown to be a general case of several bivariate distributions and is easily "transformed" under the Mellin transformation. This distribution, called the bivariate H-function distribution, includes as special cases, the univariate H-function distribution, the quarter Cauchy, McKay's bivariate gamma, a bivariate beta, and a general class of bivariate exponential distributions.
Theorems are presented to show that the products or quotients of dependent H-function variates result in an H-function variate using double Mellin transform techniques.