On the Distribution of the Product and the Sum of Generalized Shifted Gamma Random Variables

TL;DR

This paper introduces a new special function to derive closed-form probability density functions for sums and products of shifted generalized gamma variables, addressing limitations of traditional integral transforms.

Contribution

The paper develops a novel generalized Mellin transform-based function, enabling closed-form solutions for sums and products of shifted generalized gamma variables.

Findings

New function $\widehat{I}$ defined and applied

Closed-form PDFs derived for sums and products

Applications demonstrated in scientific contexts

Abstract

In general, while obtaining the probability density function of sums and products of shifted random variables, ordinary analytical methods such as Fourier and Mellin transforms tend to provide integrals which cannot be expressed in terms of ordinary Meijer G and H functions. This way, the need of defining new functions which easily enable one to write such integrals in a closed-form is inherent to the development of this area of statistical sciences. By generalizing the Mellin transform which defines the H function, a new function is established. A direct application of the so-called $\widehat{I}$ is discussed while developing the probability density function of the sum and the product of shifted generalized gamma random variables. Important special cases of the $\widehat{I}$ and their applications in science are also discussed in order to show the applicability of the function hereby defined.