Distributions of Sum, Difference, Product and Quotient of Independent Non-central Beta Type 3 Variables
Daya K. Nagar *
Instituto de Matem´ aticas Universidad de Antioquia, Calle 67, No. 53–108, Medell´Ä±n, Colombia
Yeison Arley Ramirez-Vanegas
Instituto de Matem´ aticas Universidad de Antioquia, Calle 67, No. 53–108, Medell´Ä±n, Colombia
*Author to whom correspondence should be addressed.
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.
Keywords: Beta distribution, First hypergeometric function of Appell, Gauss hypergeometric function, Non-central distribution, transformation