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

Abstract

ABSTRACT: Let X1, X2 and X3 be independent random variables, X1 and X2 having a confluent hypergeometric function kind 1 distribution with probability density function proportional to x νi−1 i 1F1(αi; βi; −xi), i = 1, 2, and X3 having a standard gamma distribution with shape parameter ν3. Define (Y1, Y2) = (X1/X3, X2/X3) and (Z1, Z2) = (X1, X2)/(X1 + X2 + X3). In this article, we derive probability density functions of (Y1, Y2) and (Z1, Z2), and study their properties. We use the second hypergeometric function of Appell to express these density functions.