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Probability distribution
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
[
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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
![{\displaystyle f(x;\nu )={\frac {2^{-\nu /2}}{\Gamma (\nu /2)}}\,x^{-\nu /2-1}e^{-1/(2x)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67f15ba6782145ec9fe605949e334f7e54f69382)
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
[
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]
- chi-squared
: If
and
, then
![{\displaystyle Y\thicksim {\text{Inv-}}\chi ^{2}(\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0756547021368d5f70277f1da7bca903de981c9)
- scaled-inverse chi-squared
: If
, then
![{\displaystyle X\thicksim {\text{inv-}}\chi ^{2}(\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1e49607e8fe408352aa80fc37b7010d3e1ceab)
- Inverse gamma
with
and
![{\displaystyle \beta ={\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e095c465b9916949fcdc94207393c4e8d9b4af)
- Inverse chi-squared distribution is a special case of type 5
Pearson distribution
See also
[
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References
[
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]
- ^
a
b
Bernardo, J.M.; Smith, A.F.M. (1993)
Bayesian Theory
, Wiley (pages 119, 431)
ISBN
0-471-49464-X
External links
[
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]
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Discrete
univariate
| with finite
support
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with infinite
support
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Continuous
univariate
| supported on a
bounded interval
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supported on a
semi-infinite
interval
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supported
on the whole
real line
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with support
whose type varies
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Mixed
univariate
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Multivariate
(joint)
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Directional
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Degenerate
and
singular
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Families
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