An Application of Dirac’s Delta Function in Bivariate Beta-Gamma Distribution

The Dirac delta function $\delta$(x)is one of the most important mathematical functions that is used successfully in physics and mathematics. Several authors have used this function to derive distributions of functions of random variables. This article consists of two main axes, namely, the use of change-of-variables technique and $\delta$(x) in deriving the ratio distribution of the two variables of bivariate Beta-Gamma distribution. The aim is to prove that $\delta$(x) is a successful alternative to the change-of-variables technique especially for complex density functions. An appropriate mathematical body of $\delta$(x) is proposed to achieve an adequate representation of the transformation task required for the target distribution. It has been proven that the two methods lead to the same result, which is that the ratio of the two variables follows a beta distribution and that $\delta$(x)is an easy and straightforward alternative to the traditional method.