Poisson(λt)

In ModelAssist we use the product λ*t as the parameter for the Poisson distribution, because it makes Poisson modeling much easier. Two examples of the Poisson distribution are shown below, with λt equal to 5 and 23 respectively:

#### Use

The Poisson(λ*t*) distribution models the number of occurrences of an event in a time t with an expected rate of l events per period t when the time between successive events follows a Poisson process (we suggest that you read the section on the Poisson process first, before continuing here).

#### Example

If *b* is the mean time between events, as used by the Exponential distribution, then λ = 1/*b*. For example, imagine that records show that a computer crashes on average once every 250 hours of operation (b=250 hours), then the rate of crashing λ is 1/250 crashes per hour. Thus a Poisson *(*1000/250*)* = Poisson(4) distribution models the number of crashes that could occur in the next 1000 hours of operation.

##### Examples in ModelAssist

The Poisson distribution is one of the most important in risk analysis, so you will find a large number of examples. Here are a few:

Time series model of events occurring randomly in time;

Bayesian simulation model to estimate herd infection;

Fire incidence modeling for integrated risk management

#### Comments

The Poisson distribution has the useful property: Poisson(*a*) + Poisson(*b*) = Poisson(*a*+*b*). This property says in words that if a accidents are expected to happen in some period and b in another period, we could estimate the variability of the total number of accidents in the total period with a Poisson(a + b).

The Poisson distribution is related to the Exponential and Gamma distributions, through the Poisson process. The Poisson distribution and process are named after the French mathematician and physicist *Siméon Denis Poisson*, though *de Moivre* (1711) derived the distribution before Poisson.

The Poisson distribution is often thought incorrectly as being applied only to rare events, perhaps because of the work by *Bortkiewicz*(1898) who looked at the frequency of infantry deaths in the Prussian Army Corps from being kicked by a horse, and who described the scenarios in which the Poisson distribution fits well as the "Law of Small Numbers". Bortkiewicz also fit Poisson distributions to child suicide rates in Prussia. But a rare event applies some subjective idea of what a 'long time' must be and the Poisson mathematics works at all scales of time.

The Excel function POISSON(x,(λ*t),0) returns the Poisson probability mass function, and POISSON(x,(λ*t),1) returns the Poisson cumulative distribution function.