This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# InverseChiSquareDistribution

 InverseChiSquareDistribution[]represents an inverse distribution with degrees of freedom. InverseChiSquareDistributionrepresents a scaled inverse distribution with degrees of freedom and scale .
• The inverse distribution InverseChiSquareDistribution[] is the distribution followed by the inverse of a -distributed random variable with degrees of freedom.
• The inverse distribution is commonly used in normal models for Bayesian data analysis.
Probability density function:
For scaled inverse distribution:
Cumulative distribution function:
For scaled inverse distribution:
Mean and variance:
Median:
Probability density function:
 Out[1]=
 Out[2]=
For scaled inverse distribution:
 Out[3]=
 Out[4]=
 Out[5]=

Cumulative distribution function:
 Out[1]=
 Out[2]=
For scaled inverse distribution:
 Out[3]=
 Out[4]=
 Out[5]=

Mean and variance:
 Out[1]=
 Out[2]=

Median:
 Out[1]=
 Scope   (7)
Generate a set of pseudorandom numbers that are inverse distributed:
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:
 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 :
Parameter influence on the CDF for each :
InverseChiSquareDistribution is closed under scaling by a positive factor:
Relationships to other distributions:
The two forms are related by a change of variable:
InverseChiSquareDistribution and ChiSquareDistribution have an inverse relationship:
InverseChiSquareDistribution is a special case of PearsonDistribution of type 5:
InverseChiSquareDistribution is not defined when either or is not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
New in 7