Distributions of Sum, Difference, Product and Quotient of Independent Non-central Beta Type 3 Variables

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 δ, respectively. Define Z = X/(X + 2Y ). Then, the random variable Z has a non-central beta type 3 distribution, Z ∼ NCB3(a, b; δ). In this article we derive density functions of sum, difference, product and quotient of two independent random variables each having noncentral beta type 3 distribution. These density functions are expressed in series involving first hypergeometric function of Appell.