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21 The Exponential Distribution

21 The Exponential DistributionFrom Discrete-Time to continuous -Time:In Chapter 6 of the text we will be considering Markov processes in con-tinuous time. In a sense, we already have a very good understanding ofcontinuous-time Markov chains based on our theory for discrete-timeMarkov chains. For example, one way to describe a continuous -timeMarkov chain is to say that it is a discrete-time Markov chain, exceptthat we explicitly model the times between transitions with contin-uous, positive-valued random variables and we explicity consider theprocess at any timet, not just at transition single most important continuous Distribution for building andunderstanding continuous -time Markov chains is the Exponential dis-tribution, for reasons which

distribution if it has probability density function f X(x|λ) = ˆ λe−λx for x>0 0 for x≤ 0, where λ>0 is called the rate of the distribution. In the study of continuous-time stochastic processes, the exponential distribution is usually used to model the time until something hap-pens in the process. The mean of the Exponential(λ ...

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