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Finding the PDF of the hypoexponential random variable using the Kad matrix similar to the general Vandermonde matrix
Authors:Khaled Smaili  Therrar Kadri
Institution:1. Department of Applied Mathematics, Faculty of Sciences, Lebanese University, Zahle, Lebanon;2. Mathematics Department, Faculty of Science, BAU, Beirut, Lebanon
Abstract:ABSTRACT

The sum of independent exponential random variables – the hypoexponential random variables – plays an important role of modeling in many domains. Khuong and Kong in (2006) Khuong, H.V., Kong, H.Y. (2006). General expression for pdf of a sum of independent exponential random variables. IEEE Commun. Lett. 10: 159161.Crossref], Web of Science ®] Google Scholar] were concerned in evaluating the performance of some diversity scheme, which deals with the problem of finding the probability density function of this hypoexponential random variable. They considered a particular case of m independent exponential random variable, when l random variables have the same mean and m ? l remaining random variables of different means and they found a closed expression of its probability density function. In this paper, we consider the general case of the hypoexponential random variable when the means do not have to be distinct. We find a more simple and general closed expression of its probability density function than that of Khuong and Kong. This expression is obtained using a new defined matrix called the Kad matrix, which is similar to the general Vandermonde matrix. Eventually, we present an application illustrating our work.
Keywords:Block martices  Erlang distribution  General Vandermonde matrix  Hypoexponential distribution  Laplace transform  Probability density function  
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