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

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.