# InverseChiSquareDistribution

represents an inverse distribution with ν degrees of freedom.

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

# Details • The inverse distribution is the distribution followed by the inverse of a -distributed random variable with ν degrees of freedom.
• The quantity follows a scaled inverse distribution where follows a distribution with ν degrees of freedom. »
• The inverse distribution is commonly used in normal models for Bayesian data analysis.
• InverseChiSquareDistribution allows ν and ξ to be any positive real numbers.
• InverseChiSquareDistribution allows ξ to be a quantity of any unit dimension and ν to be a dimensionless quantity. »
• InverseChiSquareDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Background & Context

• represents a continuous statistical distribution defined over the interval , parametrized by positive values ν and ξ indicating the degrees of freedom and the scale, respectively, of the distribution, and known as a scaled inverse chi-squared (or inverse ) distribution. Overall, the probability density function (PDF) of a scaled inverse distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its spread, and its concentration near the axis) is determined by the values of ν and ξ. In addition, the PDF of a scaled inverse distribution has tails which are "fat" in the sense that the PDF decreases algebraically rather than decreasing exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The one-parameter form (which is equivalent to InverseChiSquareDistribution[ν,1/ν]) is sometimes referred to as the unscaled inverse distribution but is most typically "the" inverse distribution.
• is the distribution followed by the reciprocal of a chi-squared random variable, whereas is the distribution followed by the -scaled reciprocal of such a random variable. In other words, if is a random variable and (where denotes "is distributed as"), then and . In Bayesian probability, the inverse chi-squared distribution is used as both a prior and posterior distribution in inferencing of normally distributed data whose variance is unknown, and the distribution is also used as a statistical foundation for Fisher's method in classification theory. InverseChiSquareDistribution is also used as a tool in various areas including Monte Carlo theory, financial mathematics, quantitative psychology, and electrical engineering.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from an inverse chi-square distribution. Distributed[x,InverseChiSquareDistribution[ν,ξ]], written more concisely as xInverseChiSquareDistribution[ν,ξ], can be used to assert that a random variable x is distributed according to an inverse chi-square distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions for inverse chi-square distributions may be given using PDF[InverseChiSquareDistribution[ν,ξ],x] and CDF[InverseChiSquareDistribution[ν,ξ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with an inverse chi-square distribution, EstimatedDistribution to estimate an inverse chi-square parametric distribution from given data, and FindDistributionParameters to fit data to an inverse chi-square distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic inverse chi-square distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic inverse chi-square distribution.
• TransformedDistribution can be used to represent a transformed inverse chi-square distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain an inverse chi-square distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving inverse chi-square distributions.
• InverseChiSquareDistribution is closely related to a number of other distributions. For example, several distributions, including GammaDistribution, ExponentialDistribution, ChiSquareDistribution, UniformDistribution, and LaplaceDistribution, can be obtained by transformations of InverseChiSquareDistribution, while NormalDistribution and FRatioDistribution are limiting values for transformed versions of InverseChiSquareDistribution. Moreover, InverseChiSquareDistribution can be viewed as a special case of PearsonDistribution and InverseGammaDistribution, and as a transformed special case of distributions including RayleighDistribution, MaxwellDistribution, and ParetoDistribution. InverseChiSquareDistribution is also closely related to BetaDistribution, StudentTDistribution, UniformDistribution, and NoncentralChiSquareDistribution.

# 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:

Closed form for symbolic order:

Closed form for symbolic order:

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: 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: