Contents

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 prob.tLocationScaleDistribution object to data using fitdist or the Distribution Fitting app, disttool.

Probability Density Function

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

Γ(ν+12)σνπΓ(ν2)[ν+(xμσ)2ν](ν+12)

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. Alternatively, you can create a prob.tLocationScaleDistribution object using fitdist or makedist, then use the pdf method to work with the object.

Cumulative Distribution Function

To compute the probability density function, use cdf. Alternatively, you can create a prob.tLocationScaleDistribution object using fitdist or makedist, then use the cdf method to work with the object.

Descriptive Statistics

The mean of the t location-scale distribution is

mean=μ,

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

var=σ2νν2,

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 prob.tLocationScaleDistribution object using fitdist or makedist. You can also use the Distribution Fitting app, disttool.

Relationship to Other Distributions

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

xμσ

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

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