Inverse-chi-squared distribution
Probability density function | |||
Cumulative distribution function | |||
Parameters | |||
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Support | |||
CDF | |||
Mean | for | ||
Median | |||
Mode | |||
Variance | for | ||
Skewness | for | ||
Excess kurtosis | for | ||
Entropy |
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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 arises in Bayesian inference, where it can be used as the prior and posterior distribution for an unknown variance of the normal distribution.
Definition
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 .
Related distributions
- 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
References
- ^ a b Bernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory, Wiley (pages 119, 431) ISBN 0-471-49464-X
External links
- InvChisquare in geoR package for the R Language.
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