Inverse-chi-square distribution

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Inverse-chi-square
Probability density function
File:Inverse chi squared density.png
Cumulative distribution function
File:Inverse chi squared distribution.png
Parameters
Support
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean for
Median
Mode
Variance for
Skewness for
Excess kurtosis for
Entropy

Moment-generating function (mgf)
Characteristic function

In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-square distribution. It is also often defined as the distribution of a random variable whose reciprocal divided by its degrees of freedom is a chi-square distribution. That is, if has the chi-square distribution with degrees of freedom, then according to the first definition, has the inverse-chi-square distribution with degrees of freedom; while according to the second definition, has the inverse-chi-square distribution with degrees of freedom.

This distribution arises in Bayesian statistics.

It is a continuous distribution with a probability density function. The first definition yields a density function

The second definition yields a density function

In both cases, and is the degrees of freedom parameter. This article will deal with the first definition only. Both definitions are special cases of the scale-inverse-chi-square distribution. For the first definition and for the second definition .

Related distributions

  • chi-square: If and then .
  • Inverse gamma with and

See also


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