---
title: "Poisson"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575214/Poisson"
format: markdown
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Poisson(<span style="color: #222222">λ</span>t)

[Poisson equations](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575986/)

  


  


In ModelAssist we use the product <span style="color: #222222">λ</span>*t as the parameter for the Poisson distribution, because it makes Poisson modeling much [easier](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574980). Two examples of the Poisson distribution are shown below, with <span style="color: #222222">λ</span>t equal to 5 and 23 respectively:

  


![image](media://5d54e417-a270-42dc-9da5-3af5617a04b0)

  


#### Use

The Poisson(<span style="color: #222222">λ</span>*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](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574979/) first, before continuing here).

  


#### Example

If *b* is the mean time between events, as used by the [Exponential](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575227/) distribution, then <span style="color: #222222">λ</span> = 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 <span style="color: #222222">λ</span> 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](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575447/);

[Bayesian simulation model to estimate herd infection](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575400/);

[Fire incidence modeling for integrated risk management](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575611/)

  


#### 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](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575227/) and [Gamma](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575231/) 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,(<span style="color: #222222">λ</span>*t),0) returns the Poisson probability mass function, and POISSON(x,(<span style="color: #222222">λ</span>*t),1) returns the Poisson cumulative distribution function.

  


  


  


  


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