Sums of Exponential Random Variables

Abstract

Exponential distribution is one of the continuous probability distributions that in most cases has been used in the analysis of Poisson processes and is the most widely used in statistical studies of reliability applications. In this project we are aiming at computing the sums of exponential random variables that have found a wide range of applications in real life mathematical modelling. In many processes involving waiting time of services, exponential distributions plays a signi cant role in making responsible statistical inferences for signi cant system output. In this study we have constructed di erent distributions for the sums of exponential random variables considering various cases where the parameter rates may be independent and identical or distinct. The generalization of the sums of exponential random variables with independent and identical parameter describes the intervals until n counts occur in the Poisson process. This forms an Erlang random variable as well proved in this project as well as hypo-exponential random variable for the case of independent and distinct parameters. The results obtained indicates variation e ects depending on the sample size of the distribution and nature of parameter rates on the e ciency of the estimation techniques chosen in comparing respective outputs in applications. Estimation of properties is determined using the method of moments and maximum likelihood estimation for some cases attempted. Owing to the relationship of exponential distribution to Poisson process, a study on the compound mixed Poisson distribution have also been provided. We have also considered to derive the probability density functions for hypo-exponential distribution for the general cases where the model parameters form arithmetic and geometric sequences.