# Documentation

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# prob.tLocationScaleDistribution class

Package: prob
Superclasses: prob.ToolboxFittableParametricDistribution

t Location-Scale probability distribution object

## Description

prob.tLocationScaleDistribution is an object consisting of parameters, a model description, and sample data for a t location-scale 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('tLocationScale') creates a t location-scale probability distribution object using the default parameter values.

pd = makedist('tLocationScale','mu',mu,'sigma',sigma,'nu',nu) creates a t location-scale probability distribution object using the specified parameter values.

### Input Arguments

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Location parameter for the t location-scale distribution, specified as a scalar value.

Data Types: single | double

Scale parameter for the t location-scale distribution, specified as a positive scalar value.

Data Types: single | double

Degrees of freedom for the t location-scale distribution, specified as a positive scalar value.

Data Types: single | double

## Properties

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Location parameter of the t location-scale distribution, stored as a scalar value.

Data Types: single | double

Scale parameter of the t location-scale distribution, stored as a positive scalar value.

Data Types: single | double

Degrees of freedom of the t location-scale distribution, stored as a positive 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

### t Location-Scale Distribution

The t location-scale distribution is useful for modeling data distributions with heavier tails (more prone to outliers) than the normal distribution. It approaches the normal distribution as ν approaches infinity, and smaller values of ν yield heavier tails.

The t location-scale distribution uses the following parameters.

ParameterDescriptionSupport
muLocation parameter$-\infty <\mu <\infty$
sigmaScale parameter$\sigma >0$
nuShape parameter$\nu >0$

The probability density function (pdf) is

$f\left(x|\mu ,\sigma ,\nu \right)=\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\sigma \sqrt{\nu \pi }\Gamma \left(\frac{\nu }{2}\right)}{\left[\frac{\nu +{\left(\frac{x-\mu }{\sigma }\right)}^{2}}{\nu }\right]}^{-\left(\frac{\nu +1}{2}\right)}\text{ };\text{ }-\infty

where $\Gamma \left(\cdot \right)$ is the Gamma function.

## Examples

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Create a t location scale distribution object using the default parameter values.

pd = makedist('tLocationScale')
pd =

tLocationScaleDistribution

t Location-Scale distribution
mu = 0
sigma = 1
nu = 5

Create a t location-scale distribution object by specifying the parameter values.

pd = makedist('tLocationScale','mu',-2,'sigma',1,'nu',20)
pd =

tLocationScaleDistribution

t Location-Scale distribution
mu = -2
sigma =  1
nu = 20

Compute the interquartile range of the distribution.

r = iqr(pd)
r =

1.3739