The Hypoexponential distribution is the distribution of the sum of n ≥ 2 independent Exponential random variables. This distribution is used in moduling multiple exponential stages in series. This distribution can be used in many domains of application. In this paper we consider the case of n exponential Random Variable having distinct parameters. Using convolution, some properties of Laplace transform and the moment generating function, we analyse this case and give new properties and identities. Moreover, we shall study particular cases when i  are arithmetic and geometric.

The sum of random variables are of interest in many areas of the sciences. In teletraffic analysis, the sum of Hyperexponential distribution is used as a model for the holding time distribution. Many authors examined this model and discussed its probability density function. In this paper, we consider the sum of independent Hyper-Erlang distributions. We showed that the probability density function of this distribution is related to probability density function of the sum of independent Erlang distributions-the Hypoexponential distribution. As a consequence, we find an exact closed expressions for the probability density function of both distribution, which are related to the Kummer confluent hypergeometric function.

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