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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

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 $X$ follows a chi-squared distribution with $\nu$ degrees of freedom then $1/X$ follows the inverse chi-squared distribution with $\nu$ degrees of freedom.

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

f(x;\nu )={\frac {2^{-\nu /2}}{\Gamma (\nu /2)}}\,x^{-\nu /2-1}e^{-1/(2x)}

In the above $x>0$ and $\nu$ is the degrees of freedom parameter. Further, $\Gamma$ is the gamma function.

The inverse chi-squared distribution is a special case of the inverse-gamma distribution. with shape parameter $\alpha ={\frac {\nu }{2}}$ and scale parameter $\beta ={\frac {1}{2}}$ .

Related distributions

chi-squared: If $X\thicksim \chi ^{2}(\nu )$ and $Y={\frac {1}{X}}$ , then $Y\thicksim {\text{Inv-}}\chi ^{2}(\nu )$
scaled-inverse chi-squared: If $X\thicksim {\text{Scale-inv-}}\chi ^{2}(\nu ,1/\nu )\,$ , then $X\thicksim {\text{inv-}}\chi ^{2}(\nu )$
Inverse gamma with $\alpha ={\frac {\nu }{2}}$ and $\beta ={\frac {1}{2}}$
Inverse chi-squared distribution is a special case of type 5 Pearson distribution

References

^ ^a ^b Bernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory, Wiley (pages 119, 431) ISBN 0-471-49464-X
^ 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.

External links

InvChisquare in geoR package for the R Language.

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda