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 $x_1^{{\nu }_1-1}{x}_2^{{\nu }_2-1}Â{(1+x_1+Âx_2)}^{-({\nu }_1+{\nu }_2+\gamma )}$\linebreak $Â\text{leftidx}{}_{2}F{}_1Â(\alpha ,\beta ;\gamma ;(1+x_1+x_2{)}^{-1})$, $x_1>0$, $x_2>0$, 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  $ÂX_1{X}_2$. 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.