# Documentation

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# prob.PoissonDistribution class

Package: prob
Superclasses: prob.ToolboxFittableParametricDistribution

Poisson probability distribution object

## Description

prob.PoissonDistribution is an object consisting of parameters, a model description, and sample data for a Poisson probability distribution.

Create a probability distribution object with specified parameter values using makedist. Alternatively, fit a distribution to data using fitdist or the Distribution Fitting app.

## Construction

pd = makedist('Poisson') creates a Poisson probability distribution object using the default parameter values.

pd = makedist('Poisson','lambda',lambda) creates a Poisson distribution object using the specified parameter value.

### Input Arguments

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Mean of the Poisson distribution, specified as a nonnegative scalar value.

Data Types: single | double

## Properties

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Mean of the Poisson distribution, stored as a nonnegative scalar value.

Data Types: single | double

Probability distribution name, stored as a character vector. This property is read-only.

Data Types: char

Data used for distribution fitting, stored as a structure containing the following:

• data: Data vector used for distribution fitting.

• cens: Censoring vector, or empty if none.

• freq: Frequency vector, or empty if none.

Data Types: struct

Logical flag for truncated distribution, stored as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated. This property is read-only.

Data Types: logical

Number of parameters for the probability distribution, stored as a positive integer value. This property is read-only.

Data Types: single | double

Covariance matrix of the parameter estimates, stored as a p-by-p matrix, where p is the number of parameters in the distribution. The (i,j) element is the covariance between the estimates of the ith parameter and the jth parameter. The (i,i) element is the estimated variance of the ith parameter. If parameter i is fixed rather than estimated by fitting the distribution to data, then the (i,i) elements of the covariance matrix are 0. This property is read-only.

Data Types: single | double

Distribution parameter descriptions, stored as a cell array of character vectors. Each cell contains a short description of one distribution parameter. This property is read-only.

Data Types: char

Logical flag for fixed parameters, stored as an array of logical values. If 0, the corresponding parameter in the ParameterNames array is not fixed. If 1, the corresponding parameter in the ParameterNames array is fixed. This property is read-only.

Data Types: logical

Distribution parameter names, stored as a cell array of character vectors. This property is read-only.

Data Types: char

Distribution parameter values, stored as a vector. This property is read-only.

Data Types: single | double

Truncation interval for the probability distribution, stored as a vector containing the lower and upper truncation boundaries. This property is read-only.

Data Types: single | double

## Methods

### Inherited Methods

 cdf Cumulative distribution function of probability distribution object icdf Inverse cumulative distribution function of probability distribution object iqr Interquartile range of probability distribution object median Median of probability distribution object pdf Probability density function of probability distribution object random Generate random numbers from probability distribution object truncate Truncate probability distribution object
 mean Mean of probability distribution object negloglik Negative log likelihood of probability distribution object paramci Confidence intervals for probability distribution parameters proflik Profile likelihood function for probability distribution object std Standard deviation of probability distribution object var Variance of probability distribution object

## Definitions

### Poisson Distribution

The Poisson distribution is appropriate for applications that involve counting the number of times a random event occurs in a given amount of time, distance, area, etc. If the number of counts follows the Poisson distribution, then the interval between individual counts follows the exponential distribution.

The Poisson distribution uses the following parameters.

ParameterDescriptionSupport
lambdaMean$\lambda \ge 0$

The probability density function of the Poisson distribution is

$f\left(x|\lambda \right)=\frac{{\lambda }^{x}}{x!}{e}^{-\lambda }\text{ };\text{ }x=0,1,2,...,\infty \text{\hspace{0.17em}}.$

## Examples

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Create a Poisson distribution object using the default parameter values.

pd = makedist('Poisson')
pd =

PoissonDistribution

Poisson distribution
lambda = 1

Create a Poisson distribution object by specifying the parameter values.

pd = makedist('Poisson','lambda',5)
pd =

PoissonDistribution

Poisson distribution
lambda = 5

Compute the variance of the distribution.

v = var(pd)
v =

5

For the Poisson distribution, both the mean and variance are equal to the parameter lambda.