Generalized bivariate beta distributions involving Appell’s hypergeometric function of the second kind

Let X1,X2X1,X2 and X3X3 be independent random variables, X1X1 and X2X2 having a confluent hypergeometric function kind 1 distribution with probability density function proportional to xiνi−11F1(αi;βi;−xi), i=1,2i=1,2, and X3X3 having a standard gamma distribution with shape parameter ν3ν3. Define (Y1,Y2)=(X1/X3,X2/X3)(Y1,Y2)=(X1/X3,X2/X3) and (Z1,Z2)=(X1,X2)/(X1+X2+X3)(Z1,Z2)=(X1,X2)/(X1+X2+X3). In this article, we derive probability density functions of (Y1,Y2)(Y1,Y2) and (Z1,Z2)(Z1,Z2), and study their properties. We use the second hypergeometric function of Appell to express these density functions.