The Poisson and Exponential Distributions

1 Published by theApplied Probability Trust Applied Probability Trust 2005123 The Poisson andExponential DistributionsJOHN C. B. COOPER1. IntroductionThe Poisson distribution is a discrete distribution with probability mass functionP(x)=e xx!,wherex=0,1,2,.., the mean of the distribution is denoted by , and e is the variance of this distribution is also equal to .The Exponential distribution is a continuous distribution with probability density functionf(t)= e t,wheret 0 and the parameter >0. The mean and standard deviation of this distributionare both equal to 1/ .The cumulative Exponential distribution isF(t)= 0 e tdt=1 e t.(1)2. Relation between the Poisson and Exponential distributionsAn interesting feature of these two Distributions is that, if the Poisson provides an appropriatedescription of the number of occurrences per interval of time, then the Exponential will providea description of the length of time between occurrences.

2 To understand this, consider that, ina Poisson process, if events occur on average at the rate of per unit of time, then there willbe on average toccurrences pertunits of time. The Poisson distribution describing thisprocess is thereforeP(x)=e t( t)x/x!, from whichP(x=0)=e tis the probability ofno occurrences intunits of interpretation ofP(x=0)=e tis that this is the probability that the time,T,to the first occurrence is greater thant, (T > t)=P(x=0| = t)=e , the probability that an event does occur duringtunits of time is given byP(T t)=1 P(x=0| = t)=1 e that this is the cumulative Exponential distribution which, when differentiated with respecttot, produces the probability density function of the Exponential distributionf(t)= e t.