## The Poisson Distribution

The Poisson distribution was introduced briefly in Chapter 10, Section 10.5.5, which dealt with the issue of software reliability. The Poisson is a discrete distribution given by the following formula:

where P(k) is the probability of exactly k events of interest, X is the rate at which such events are occurring, and t is the time (or space) over which the events are occurring. This distribution may be used in situations for which events happen at some rate and we wish to ascertain the probability of some number of events occurring in a total period of time, or in a certain space.

Example. Cars are passing a toll booth at an overall rate of about 120 cars per hour. The probability that exactly three cars will pass through the toll booth in a period of 1 minute would be p(3) = [(2)(1)]3 exp [ (2)(1)] = 018

The Poisson distribution, when k = 0, reduces to P(0) = exp (- Xt), which is the exponential distribution. For example, if failures were occurring in a system at a rate of X, and we were concerned with the likelihood of having no failures in some period of time, t, we would use the exponential to make this calculation. The subject of reliability is examined again later in this chapter.

11.5 the normal (gaussian) distribution 325

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