On the distribution of a random variable involved in an independent ratio

TL;DR

This paper derives the exact distribution of a random variable involved in a ratio using inverse integral transforms, introduces new distributions, and provides several illustrative examples.

Contribution

It presents a novel method to find the distribution of a ratio-involved variable and introduces new distributions based on this approach.

Findings

Derived exact distribution of the ratio-involved variable

Introduced new distributions from the ratio analysis

Provided multiple examples demonstrating the method

Abstract

In this paper, using inverse integral transforms, we derive the exact distribution of the random variable $X$ that is involved in the ratio $Z \stackrel{d}{=} X/(X+Y)$ where $X$ and $Y$ are independent random variables having the same support, and $Z$ and $Y$ have known distributions. We introduce new distributions this way. As applications of the obtained results, several examples are presented.