t Location-Scale Distribution

Overview

Thetlocation-scale distribution is useful for modeling data distributions with heavier tails (more prone to outliers) than the normal distribution. It approaches the normal distribution asνapproaches infinity, and smaller values ofνyield heavier tails.

Parameters

Thetlocation-scale distribution uses the following parameters.

Parameter	Description	金宝app
μ	Location parameter	–∞ < μ < ∞
σ	Scale parameter	σ > 0
ν	Shape parameter	ν > 0

To estimate distribution parameters, usemle. Alternatively, fit atLocationScaleDistributionobject to data usingfitdistor theDistribution Fitter应用程序。

Probability Density Function

The probability density function (pdf) of thetlocation-scale distribution is

$\frac{Γ (\frac{ν + 1}{2})}{σ \sqrt{ν π} Γ (\frac{ν}{2})} {[\frac{ν + {(\frac{x - μ}{σ})}^{2}}{ν}]}^{- (\frac{ν + 1}{2})}$

where Γ( • ) is the gamma function,µis the location parameter,σis the scale parameter, andνis the shape parameter .

To compute the probability density function, usepdfand specify'tLocationScale'. Alternatively, you can create atLocationScaleDistributionobject usingfitdistormakedist, then use thepdfto work with the object.

Cumulative Distribution Function

To compute the probability density function, usecdfand specify'tLocationScale'. Alternatively, you can create atLocationScaleDistributionobject usingfitdistormakedist, then use thecdfto work with the object.

Descriptive Statistics

The mean of thetlocation-scale distribution is

$mean = μ,$

whereμis the location parameter. The mean is only defined for shape parameter valuesν> 1. For other values ofν, the mean is undefined.

The variance of thetlocation-scale distribution is

$var = σ^{2} \frac{ν}{ν - 2},$

whereμis the location parameter andνis the shape parameter. The variance is only defined for values ofν> 2. For other values ofν, the variance is undefined.

To compute the mean and variance, create atLocationScaleDistributionobject usingfitdistormakedist. You can also use theDistribution Fitter应用程序。

Relationship to Other Distributions

Ifxhas atlocation-scale distribution, with parametersµ,σ, andν, then

$\frac{x - μ}{σ}$

has a Student'stdistribution withνdegrees of freedom.