InverseChiSquareDistribution

InverseChiSquareDistribution[ν]

represents an inverse distribution with ν degrees of freedom.

InverseChiSquareDistribution[ν,ξ]

represents a scaled inverse distribution with ν degrees of freedom and scale ξ.

Details

Background & Context

Examples

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Basic Examples  (4)

Probability density function:

For scaled inverse distribution:

Cumulative distribution function:

For scaled inverse distribution:

Mean and variance:

Median:

Scope  (8)

Generate a sample of pseudorandom numbers from an inverse distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:

Skewness depends only on the number of degrees of freedom:

In the limit, the distribution becomes symmetric:

Kurtosis depends only on the number of degrees of freedom:

In the limit, kurtosis is the same as for NormalDistribution:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

Closed form for symbolic order:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

For scaled inverse distribution:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the mean size:

Applications  (2)

The posterior distribution of variance of a normal distribution with zero mean was found to be InverseChiSquareDistribution with parameters and scale . Find the most likely value of the variance:

Find the expected variance:

InverseChiSquareDistribution is a conjugate prior for the likelihood of normal distribution with known mean and unknown variance:

Update prior using data sample:

Posterior distribution is again InverseChiSquareDistribution with new parameters and :

Properties & Relations  (7)

InverseChiSquareDistribution is closed under scaling by a positive factor:

Relationships to other distributions:

InverseChiSquareDistribution[ν] has scale :

The two forms are related by a change of variable:

InverseChiSquareDistribution is a special case of InverseGammaDistribution:

InverseChiSquareDistribution and ChiSquareDistribution have an inverse relationship:

InverseChiSquareDistribution is a special case of type 5 PearsonDistribution:

InverseChiSquareDistribution is a special case of PearsonDistribution of type 5:

Possible Issues  (2)

InverseChiSquareDistribution is not defined when either ν or ξ is not a positive real number:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

Neat Examples  (1)

PDFs for different ν values with CDF contours:

Wolfram Research (2008), InverseChiSquareDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseChiSquareDistribution.html (updated 2016).

Text

Wolfram Research (2008), InverseChiSquareDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseChiSquareDistribution.html (updated 2016).

BibTeX

@misc{reference.wolfram_2021_inversechisquaredistribution, author="Wolfram Research", title="{InverseChiSquareDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/InverseChiSquareDistribution.html}", note=[Accessed: 20-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_inversechisquaredistribution, organization={Wolfram Research}, title={InverseChiSquareDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/InverseChiSquareDistribution.html}, note=[Accessed: 20-October-2021 ]}

CMS

Wolfram Language. 2008. "InverseChiSquareDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/InverseChiSquareDistribution.html.

APA

Wolfram Language. (2008). InverseChiSquareDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseChiSquareDistribution.html