In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distributionBernardo, J.M.; Smith, A.F.M. (1993) ''Bayesian Theory'' ,Wiley (pages 119, 431) ) is a

continuous probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

of a positive-valued random variable. It is closely related to the

chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...

. It arises in

Bayesian inference Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, a ...

, where it can be used as the

prior Prior (or prioress) is an ecclesiastical title for a superior in some religious orders. The word is derived from the Latin for "earlier" or "first". Its earlier generic usage referred to any monastic superior. In abbeys, a prior would be l ...

and

posterior distribution The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...

for an unknown

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...

of the

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

Definition

The inverse-chi-squared distribution (or inverted-chi-square distribution ) is the

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

of a random variable whose

multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...

(reciprocal) has a

. It is also often defined as the distribution of a random variable whose reciprocal divided by its degrees of freedom is a chi-squared distribution. That is, if

X

has the chi-squared distribution with

\nu

degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

, then according to the first definition,

1/X

has the inverse-chi-squared distribution with

\nu

degrees of freedom; while according to the second definition,

\nu/X

has the inverse-chi-squared distribution with

\nu

degrees of freedom. Information associated with the first definition is depicted on the right side of the page. The first definition yields a

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

given by :

f_1(x; \nu) = \frac\,x^  e^,

while the second definition yields the density function :

f_2(x; \nu) =
\frac  x^  e^ .

In both cases,

x>0

and

\nu

is the

parameter. Further,

\Gamma

is the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

. Both definitions are special cases of the

scaled-inverse-chi-squared distribution The scaled inverse chi-squared distribution is the distribution for ''x'' = 1/''s''2, where ''s''2 is a sample mean of the squares of ν independent normal random variables that have mean 0 and inverse variance 1/σ2 = τ2. The distribu ...

. For the first definition the variance of the distribution is

\sigma^2=1/\nu ,

while for the second definition

\sigma^2=1

Related distributions

* chi-squared: If

X \thicksim \chi^2(\nu)

and

Y = \frac

, then

Y \thicksim \text\chi^2(\nu)

* scaled-inverse chi-squared: If

X \thicksim \text\chi^2(\nu, 1/\nu) \,

, then

X \thicksim \text\chi^2(\nu)

* Inverse gamma with

\alpha = \frac

and

\beta = \frac

* Inverse chi-squared distribution is a special case of type 5

Pearson distribution The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics. History The Pearson system ...

References

External links

InvChisquare
in geoR package for the R Language. {{ProbDistributions, continuous-semi-infinite Continuous distributions Exponential family distributions Probability distributions with non-finite variance

Definition

Related distributions

See also

References

External links