Non-Central Beta Type 3 Distribution

Abstract

Let $X$ and $Y$ be independent random variables, $X$ having a gamma distribution with  shape parameter  $a$ and $Y$ having a non-central
gamma distribution with shape and non-centrality parameters $b$ and
$\delta$, respectively. Define $ W ={X}/(X + 2 Y)$. Then, the random
variable $W$ has a non-central beta type 3 distribution, $W\sim
\textnormal{NCB3} (a,b;\delta)$. In this article we study several of its
properties. We also give a multivariate generalization of the
non-central beta type 3 distribution and derive its properties.