Poisson Distribution

The Poisson distribution, also known as the Poissonian distribution, models the number of occurrences of an event in a fixed interval of time or space where each event has a constant probability of occurring. It is closely related to the Exponential Distribution, which models the time between successive occurrences.

The Poisson distribution has one parameter $\mu$ ('mu'), which specifies the mean number of occurrences per unit time. The probability mass function (PMF) is given by:

$$P(X=k)=\frac{{\mu }^k{e}^{-\mu }}{k!}$$

The cumulative distribution function (CDF) is:

$$F(k;\mu )=e^{-\mu }\sum _{i=0}^k\frac{{\mu }^i}{i!}$$

The domain of the Poisson distribution is $k\in \{0,1,2,\dots \}$. The parameter $\mu$ must satisfy $\mu >0$.

The Poisson distribution is widely used in various fields due to its ability to model the number of occurrences of rare events. Common applications include:

• The number of cars passing a road that is not too busy.

• The number of failures of a piece of equipment that is replaced with identical copies when it fails.

The Poisson distribution has several important statistical properties:

The Poisson distribution is closely related to several other distributions:

• The Binomial Distribution can be approximated by the Poisson distribution when the number of trials is very large and the probability of success is very small.

• The Exponential Distribution models the time between successive occurrences of events that follow a Poisson process.

The Poisson distribution is implemented by the PoissonDistribution class. It has one constructor which takes one argument: the mean number of events per unit time. The following constructs a Poisson distribution with mean 4.4:

var poisson = new PoissonDistribution(4.4);

Dim poisson = New PoissonDistribution(4.4)

No code example is currently available or this language may not be supported.

let poisson = PoissonDistribution(4.4)

The PoissonDistribution class has no specific properties. The mean number of events per unit time is returned by the Mean property.

PoissonDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator.

var random = new Pcg32();
int sample = PoissonDistribution.Sample(random, 4.4);

Dim random = New Pcg32()
Dim sample = PoissonDistribution.Sample(random, 4.4)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = PoissonDistribution.Sample(random, 4.4)

The above example uses the Pcg32 class to generate uniform random numbers.

For details of the properties and methods common to all discrete probability distribution classes, see the topic on Discrete Probability Distributions.

• "Introduction to Probability Models" by Sheldon M. Ross.

• "Probability and Statistics" by Morris H. DeGroot and Mark J. Schervish.

Other Resources