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Inverse-chi-squared distribution

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Inverse-chi-squared
Probability density function
Cumulative distribution function
Parameters
Support
PDF
CDF
Mean for
Median
Mode
Variance for
Skewness for
Excess kurtosis for
Entropy

MGF ; does not exist as real valued function
CF

In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution[1]) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution. It is used in Bayesian inference as conjugate prior for the variance of the normal distribution.[2]

Definition

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The inverse chi-squared distribution (or inverted-chi-square distribution[1] ) is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-squared distribution.

If follows a chi-squared distribution with degrees of freedom then follows the inverse chi-squared distribution with degrees of freedom.

The probability density function of the inverse chi-squared distribution is given by

In the above and is the degrees of freedom parameter. Further, is the gamma function.

The inverse chi-squared distribution is a special case of the inverse-gamma distribution. with shape parameter and scale parameter .

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  • chi-squared: If and , then
  • scaled-inverse chi-squared: If , then
  • Inverse gamma with and
  • Inverse chi-squared distribution is a special case of type 5 Pearson distribution

See also

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References

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  1. ^ a b Bernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory, Wiley (pages 119, 431) ISBN 0-471-49464-X
  2. ^ Gelman, Andrew; et al. (2014). "Normal data with a conjugate prior distribution". Bayesian Data Analysis (Third ed.). Boca Raton: CRC Press. pp. 67–68. ISBN 978-1-4398-4095-5.
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