The moment generating function of a bivariate gamma-type distribution Abdus Saboor a,⇑ , Serge B. Provost b , Munir Ahmad c a Department of Mathematics, Kohat University of Science & Technology, Kohat 26000, Pakistan b Department of Statistical & Actuarial Sciences, The University of Western Ontario, London, Canada N6A 5B7 c National College of Business Administration & Economics, 40 E/1, Gulberg-III, Lahore 54660, Pakistan article info Keywords: Bivariate distributions Generalized gamma distributions Inverse Mellin transform Moment generating function Moments abstract A bivariate gamma-type density function involving a confluent hypergeometric function of two variables is being introduced. The inverse Mellin transform technique is employed in conjunction with the transformation of variable technique to obtain its moment generating function, which is expressed in terms of generalized hypergeometric functions. Its cumu- lative distribution function is given in closed form as well. Many distributions such as the bivariate Weibull, Rayleigh, half-normal and Maxwell distributions can be obtained as limiting cases of the proposed gamma-type density function. Computable representa- tions of the moment generating functions of these distributions are also provided. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction The univariate gamma distribution plays a major role in statistics and is involved in countless applications. The first form of its bivariate extension has been introduced by McKay [13]. An interesting application of this distribution was considered by Clarke [3] in connection with the joint distribution of annual streamflow and areal precipitation. Cherian [2] proposed another form of the bivariate gamma distribution, and Prekopa and Szantai [18] utilized it to study the streamflow of a river. Kibble [10] and Moran [14] discussed a symmetrical bivariate gamma distribution whose joint characteristic function is given by 1 fð1  it 1 Þð1  it 2 Þþ w 2 t 1 t 2 g a ; a > 0; ð1:1Þ where i ¼ ffiffiffiffiffiffiffi 1 p . Asymmetrical extensions were introduced by Sarmanov [21,22], and another generalization was proposed by Jensen [8] and Smith et al. [23]. Jensen [8] also extended Moran’s bivariate gamma distributions. Recently Nadarajah and Gupta [15] introduced two bivariate gamma distributions based on a characterizing property involving products of gamma and beta random variables. They provided certain representations of their joint densities, prod- uct moments, conditional densities and moments. Some of those representations involve special functions such as the com- plementary incomplete gamma and Whittaker’s functions. The Farlie–Gumbel–Morgenstern type bivariate gamma distribution was studied by D’Este [4] and Gupta and Wong [7]. Dussauchoy and Berland [5] introduced a joint distribution in the form of a confluent hypergeometric function of two dependent gamma random variables X 1 and X 2 with the property that X 2  bX 1 and X 1 are independent. Nakhi and Kalla [17] defined a probability density function involving a generalized r-Gauss hypergeometric function and discussed its associated statistical functions. Saxena and Kalla [24] studied a new mix- ture distribution associated with the Fox–Wright generalized hypergeometric function and discussed some associated 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.05.057 ⇑ Corresponding author. E-mail addresses: saboorhangu@gmail.com, dr.abdussaboor@um.kust.edu.pk (A. Saboor). Applied Mathematics and Computation 218 (2012) 11911–11921 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc