# 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.

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### `mu` — Location parameter`0` (default) | scalar value

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

Data Types: `single` | `double`

### `sigma` — Scale parameter`1` (default) | positive scalar value

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

Data Types: `single` | `double`

### `nu` — Degrees of freedom`5` (default) | positive scalar value

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

Data Types: `single` | `double`

## Properties

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### `mu` — Location parameterscalar value

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

Data Types: `single` | `double`

### `sigma` — Scale parameterpositive scalar value

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

Data Types: `single` | `double`

### `nu` — Degrees of freedompositive scalar value

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

Data Types: `single` | `double`

### `DistributionName` — Probability distribution nameprobability distribution name string

Probability distribution name, stored as a valid probability distribution name string. This property is read-only.

Data Types: `char`

### `InputData` — Data used for distribution fittingstructure

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`

### `IsTruncated` — Logical flag for truncated distribution`0` | `1`

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`

### `NumParameters` — Number of parameterspositive integer value

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

Data Types: `single` | `double`

### `ParameterCovariance` — Covariance matrix of the parameter estimatesmatrix of scalar values

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 `i`th parameter and the `j`th parameter. The (`i`,`i`) element is the estimated variance of the `i`th 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`

### `ParameterDescription` — Distribution parameter descriptionscell array of strings

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

Data Types: `char`

### `ParameterIsFixed` — Logical flag for fixed parametersarray of logical values

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`

### `ParameterNames` — Distribution parameter namescell array of strings

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

Data Types: `char`

### `ParameterValues` — Distribution parameter valuesvector of scalar values

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

Data Types: `single` | `double`

### `Truncation` — Truncation intervalvector of scalar values

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
`mu`Location parameter$-\infty <\mu <\infty$
`sigma`Scale parameter$\sigma >0$
`nu`Shape 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 Default Parameters

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 Using Specified Parameters

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```