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# prob.NegativeBinomialDistribution class

Package: prob
Superclasses: prob.ToolboxFittableParametricDistribution

Negative binomial distribution object

## Description

prob.NegativeBinomialDistribution is an object consisting of parameters, a model description, and sample data for a negative binomial 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('NegativeBinomial') creates a negative binomial probability distribution object using the default parameter values.

pd = makedist('NegativeBinomial','R',r,'p',p) creates a negative binomial probability distribution object using the specified parameter values.

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### r — Number of successes1 (default) | positive scalar value

Number of successes for the negative binomial distribution, specified as a positive scalar value.

Data Types: single | double

### p — Probability of success0.5 (default) | positive scalar value in the range (0,1]

Probability of success of any individual trial for the negative binomial distribution, specified as a positive scalar value in the range (0,1].

Data Types: single | double

## Properties

 r Number of successes for the negative binomial distribution, stored as a positive scalar value. p Probability of success of any individual trial for the negative binomial distribution, specified as a positive scalar value in the range (0,1]. DistributionName Name of the probability distribution, stored as a valid probability distribution name string. This property is read-only. InputData 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. This property is read-only. IsTruncated 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. NumParameters Number of parameters for the probability distribution, stored as a positive integer value. This property is read-only. ParameterCovariance 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. ParameterDescription Descriptions of distribution parameters, stored as a cell array of strings. Each cell contains a short description of one distribution parameter. This property is read-only. ParameterIsFixed 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. ParameterNames Names of distribution parameters, stored as a cell array of strings. This property is read-only. ParameterValues Values of distribution parameters, stored as a vector. This property is read-only. Truncation Truncation interval for the probability distribution, stored as a vector containing the lower and upper truncation boundaries. This property is read-only.

## 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 loglikelihood 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

### Negative Binomial Distribution

The negative binomial distribution models the number of failures x before a specified number of successes is reached in a series of independent, identical trials. This distribution can also model count data, in which case r does not need to be an integer value.

The negative binomial distribution uses the following parameters.

ParameterDescriptionSupport
RNumber of successes
pProbability of success

The probability density function (pdf) when r is an integer is

where and is the binomial coefficient.

When r is not an integer, the binomial coefficient in the definition of the pdf is replaced by the equivalent expression

where is the Gamma function.

## Examples

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### Create a Negative Binomial Distribution Object Using Default Parameters

Create a negative binomial distribution object using the default parameter values.

`pd = makedist('NegativeBinomial')`
```pd =

NegativeBinomialDistribution

Negative Binomial distribution
R =   1
P = 0.5```

### Create a Negative Binomial Distribution Object Using Specified Parameters

Create a negative binomial distribution object by specifying the parameter values.

`pd = makedist('NegativeBinomial','r',5,'p',.1)`
```pd =

NegativeBinomialDistribution

Negative Binomial distribution
R =   5
P = 0.1```

Compute the mean of the distribution.

`m = mean(pd)`
```m =

45```