Poisson Distribution

The Poisson distribution is used to describe the number of events per unit of a continuum.  It is named after the 19th century French mathematician and scientist Simeon-Denis Poisson.

It is a useful modelling tool with wide and varied applications which include simulating insurance claims, the requests made to a web server and the goals scored in a soccer match.

The distribution is defined by a single parameter which is the mean number of events per unit of continuum, e.g. claims/month, hits/minute or goals/match.

When the mean is greater than 30 events, it is possible to use the normal distribution as an approximation.  This removes the need to raise numbers to large powers and calculate large factorials.

The intervals between events are exponentially distributed.

Related Topics

Normal Distribution
Exponential Distribution
The Poisson distribution in soccer

Profile

Parameters

Parameter	Description	Characteristics
mean	The mean number of events per unit of continuum.	A float > 0

Functions

Properties

If the mean is an integer, the distribution is bimodal with modes at the mean and mean minus one, otherwise, the mode is ceil (mean)

Normal Approximation

The parameters for the normal approximation are:

As the normal distribution is continuous and the Poisson discrete, the probability and cumulative functions become:

When using the Normal approximation to the Poisson in computer applications it is advisable to trap mean values less than 10, otherwise the left tail can represent negative numbers.

Example

The Poisson distribution can be applied to the hit rate on a lightly loaded web site.  The graph below shows the distribution of hits/hour on a recipe web-site before the evening feed.

Whilst the fit is not statistically significant at the 10% level, for many applications it would be an adequate model for simulating the load on a web server.

Calculation

Both Microsoft's Excel and Google's Spreadsheet Document have a Poisson function which is in the the form:

=poisson(x,mean,cumulative)

For example, to calculate the probability of 1 event (p(1)), where the mean is 2.0, use:

formula=poisson(1,2.0,false) result=0.271

To calculate the probability of the number of events being <= 1, or the cumulative probability (f(1)), then use:

formula=poisson(1,2.0,true) result=0.406

Random Number Generation

Follow the link to the section on random number generation which describes a general method for deriving random numbers from discrete distributions.

Parameter Estimation

The matching moment and maximum likelihood equations for the mean are the same.

Page Updated: 02-Mar-2008

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