The Exponential Distribution and the Application to Markov Models

This paper discusses the characteristics of the exponential distribution and the related distribution functions including gamma, weibull and lognormal then relates some of their properties to the application of Markov models. One of the major properties is forgetfulness, the consequence of this is that Markov and stationarity assumptions imply that the times between events must be negative-exponentially distributed. To make a decision on the application of Markov model to any process in real life situation, it is advised that it should be fitted to the form of the negative exponential density functions which implies that the most likely times are close to zero, and very long times are increasingly unlikely. That is, the most likely values are considered to be clustered about the mean, and large deviations from the mean are viewed as increasingly unlike. If this characteristic of the negative exponential distribution seems incompatible with the application one has in mind then a Markov model may not be appropriate. Keywords; Exponential distribution, Random variables, Memory-less, Markov models, Stationarity assumption, Application. Introduction In making mathematical models for a realworld phenomenon it is always necessary to make certain simplifying assumptions so as render the mathematics tractable. One simplifying assumption that is often made when modeling with Markov principle is to assume that certain random variables are exponentially distributed. The reason for this is that exponential distribution is both relatively easy to work it and is often a good approximation to the actual distribution Ross(1989). An important simplifying assumption in making Markov chain models is that the time it takes to make a transition (random variable) be described by negativeexponential distribution. In some of the applications Abubakar(2007) utilized both the exponential and Weibull respectively to describe the waiting time in the states of semi-Markov model for leprosy treatment. Also Abubakar(2010) considered as a random variable the time it takes for sudan savannah to be transformed to sahel savannah using Weibull distribution function in a semi-Markov model for desertification. It is therefore of interest in this paper to examine the exponential distribution function and its application to modeling Markov processes. The exponential distribution The probability density function of the random variable T having the exponential distribution is f(t) = Kohlas(1982). The distribution has as a parameter. also determines the shape of the distribution. The mean  of the exponential distribution is