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.
Publisher
University of Nairobi
Subject
Exponential Random VariablesRights
Attribution-NonCommercial-NoDerivs 3.0 United StatesUsage Rights
http://creativecommons.org/licenses/by-nc-nd/3.0/us/Collections
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