On the Distributions of Product and Quotient of two Independent $\hat{I}$-function variates

TL;DR

This paper introduces a new $\hat{I}$-function distribution that generalizes many classical distributions and explores the distribution of products and quotients of two independent $\hat{I}$-function variates.

Contribution

The paper presents a novel $\hat{I}$-function distribution and derives its properties, including the distributions of products and quotients of independent variates, unifying several classical distributions.

Findings

Product and quotient of two independent $\hat{I}$-function variates also follow the $\hat{I}$-function distribution.

The $\hat{I}$-function distribution encompasses gamma, beta, exponential, normal, H-function, and G-function distributions as special cases.

The new distribution provides a unified framework for various classical distributions.

Abstract

The study of probability distributions for random variables and their algebraic combinations has been a central focus driving the advancement of probability and statistics. Since the 1920s, the challenge of calculating the probability distributions of sums, differences, products, and quotients of independent random variables have drawn the attention of numerous statisticians and mathematicians who studied the algebraic properties and relationships of random variables. Statistical distributions are highly helpful in data science and machine learning, as they provide a range of possible values for the variables, aiding in the development of a deeper understanding of the underlying problem. In this paper, we have presented a new probability distribution based on the $\hat{I}$-function. Also, we have discussed the applications of the $\hat{I}$ function, particularly in deriving the distributions of product and the quotient involving two independent $\hat{I}$ function variates. Additionally, it has been shown that both the product and quotient of two independent $\hat{I}$-function variates also follow the $\hat{I}$-function distribution. Furthermore, the new distribution, known as the $\hat{I}$-function distribution, includes several well-known classical distributions such as the gamma, beta, exponential, normal H-function, and G-function distributions, among others, as special cases. Therefore, the $\hat{I}$-function distribution can be considered a characterization or generalization of the above-mentioned distributions.