Bivariate Generalization of the Inverted Hypergeometric Function Type I Distribution

Abstract

The bivariate  inverted hypergeometric function type I distribution
 is defined by the probability density function proportional to $ x_1^{\nu_1-1} x_2^{\nu_2-1}  \left(1 + x_{1} + x_{2}\right)^{-(\nu_1+\nu_2+\gamma)}$\linebreak $  \leftidx{_2}{F}{_1}
 (\alpha,\beta;\gamma;(1+ x_{1}+x_{2})^{-1} )$, $x_1>0$, $x_2>0$, where  $\nu_1$, $\nu_2$, $\alpha$, $\beta$ and $\gamma$ are suitable constants.  In this article, we study several properties of this distribution and derive density functions of $X_1/X_2 $, $X_1/(X_1+X_2)$,  $X_1+X_2$  and  $  X_1 X_2 $. We also consider several other  products involving bivariate  inverted hypergeometric  function type I, beta type I, beta type II, beta type III, Kummer-beta and hypergeometric function type I variables.