## t Location-Scale Distribution

### Overview

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.

### Parameters

The t location-scale distribution uses the following parameters.

ParameterDescriptionSupport
μLocation parameter–∞ < μ < ∞
σScale parameterσ > 0
νShape parameterν > 0

To estimate distribution parameters, use `mle`. Alternatively, fit a `tLocationScaleDistribution` object to data using `fitdist` or the Distribution Fitter app.

### Probability Density Function

The probability density function (pdf) of the t location-scale distribution is

`$\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\sigma \sqrt{\nu \pi }\Gamma \left(\frac{\nu }{2}\right)}{\left[\frac{\nu +{\left(\frac{x-\mu }{\sigma }\right)}^{2}}{\nu }\right]}^{-\left(\frac{\nu +1}{2}\right)}$`

where Γ( • ) is the gamma function, µ is the location parameter, σ is the scale parameter, and ν is the shape parameter .

To compute the probability density function, use `pdf` and specify `'tLocationScale'`. Alternatively, you can create a `tLocationScaleDistribution` object using `fitdist` or `makedist`, then use the `pdf` to work with the object.

### Cumulative Distribution Function

To compute the probability density function, use `cdf` and specify `'tLocationScale'`. Alternatively, you can create a `tLocationScaleDistribution` object using `fitdist` or `makedist`, then use the `cdf` to work with the object.

### Descriptive Statistics

The mean of the t location-scale distribution is

`$\text{mean}=\mu \text{\hspace{0.17em}},$`

where μ is the location parameter. The mean is only defined for shape parameter values ν > 1. For other values of ν, the mean is undefined.

The variance of the t location-scale distribution is

`$\mathrm{var}={\sigma }^{2}\frac{\nu }{\nu -2}\text{\hspace{0.17em}},$`

where μ is the location parameter and ν is the shape parameter. The variance is only defined for values of ν > 2. For other values of ν, the variance is undefined.

To compute the mean and variance, create a `tLocationScaleDistribution` object using `fitdist` or `makedist`. You can also use the Distribution Fitter app.

### Relationship to Other Distributions

If x has a t location-scale distribution, with parameters µ, σ, and ν, then

`$\frac{x-\mu }{\sigma }$`

has a Student's t distribution with ν degrees of freedom.