J. Appl. Prob. 46, 272–283 (2009) Printed in England  Applied Probability Trust 2009 ON THE SUMS OF COMPOUND NEGATIVE BINOMIAL AND GAMMA RANDOM VARIABLES P. VELLAISAMY ∗ ∗∗ and N. S. UPADHYE, ∗ Indian Institute of Technology Bombay Abstract We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability of m-out-of-n:G systems and to the shortest path problem in graph theory are also discussed. Keywords: Compound negative binomial distribution; convolution; random parameter representation; compound Poisson distribution; sums of gamma random variables; Poisson process 2000 Mathematics Subject Classification: Primary 60E05 Secondary 62E15 1. Introduction The compound negative binomial (CNB) model arises naturally in several fields, such as insurance mathematics and actuarial sciences, and has been studied by several authors. For a recent reference, see Drekic and Willmot (2005) and the references therein. It also arises in nonactuarial applications (see Johnson et al. (2005, pp. 232–250) and Vellaisamy and Upadhye (2007)). Recently, Furman (2007) studied the sums of independent negative binomial random variables and obtained an interesting recurrence relation for computing its probability mass function (PMF). He also showed that the convolution of a negative binomial distribution with arbitrary parameters is a negative binomial distribution, but with a random parameter. In Section 2 we first derive an exact expression for the distribution of sums of CNB random variables. For the negative binomial (NB) case, this expression reduces to a finite-sum expression which is numerically compared with the series expression of Furman (2007). We also obtain a simple random parameter representation for the convolution of CNB distributions, where the compounding distributions Q j = Q. Theorems 2.1 and 2.2 of Furman (2007) follow as special cases. Our approach is essentially that of Furman (2007), except that we use the distribution itself rather than using its moment generating function (MGF). If the Q j s are different then the convolution of CNB distributions is neither a CNB nor a mixture of CNBs. In such cases, a compound Poisson (CP) representation is presented. It is also shown that a sum of certain dependent CP variables is again a CP variable. It is well known Received 18 September 2008; revision received 21 January 2009. ∗ Postal address: Department of Mathematics, Indian Institute ofTechnology Bombay, Powai, Mumbai, 400 076, India. ∗∗ Email address: pv@math.iitb.ac.in 272 https://www.cambridge.org/core/terms. https://doi.org/10.1239/jap/1238592129 Downloaded from https://www.cambridge.org/core. IP address: 54.161.69.107, on 05 Jun 2020 at 14:53:00, subject to the Cambridge Core terms of use, available at