## The Poisson Distribution

The Poisson distribution models a situation in whichevents happen at a certain rate, so many accidents in this stretch ofroad per month, or so many misprints in this book per page. It iswritten ,whereisthe mean number of events in a certain time period. The Poissondistribution has the very useful feature that it is scalable so thatif you double the time period, you double the expected number ofevents. Since the Poisson distribution has only one parameter, theexpected or mean number of events,this is a very useful feature.

The Poisson distribution is defined byItmay be used as in the following examples.

Example: On a stretch of motorwayaccidents occur at a rate of 0.9 per month.

a) Show that the probability of noaccidents in the next month is 0.407, to 3 significant figures.

b) Find the probability of exactly 2 accidents occuring in thenext 6 month period.

c)Find the probability of at least two accidents in the next sixmonths.

a)to3sf.

b)In 1 month we expect 0.9 accidents, so in 6 months weexpect 6*0.9=5.4 accidents. The distribution becomesUsingthis distribution we findto4sf.

c)

to4 dp.

Sometimes two distributions are combined. Theprobability of no accidents in a month is 0.407. Suppose then we needto find the probability of having exactly 3 months in the next yearwith no accidents. The probability 0.407 is fixed. The number ofmonths n, is 12. Of course now it is a binomial distribution,Fora binomial distribution,

to4dp.

This sort of thing is actually quite common, and meansthat every situation should be analysed carefully. It is not alwaysthe case that a single distribution should be used throughout foreach question.