In probability and statistics , the inverse-chi-square distribution is the probability distribution of a random variable whose inverse has a chi-square distribution . It is also often defined as the distribution of a random variable whose inverse divided by its degrees of freedom is a chi-square distribution. That is, if
X
{\displaystyle X}
ν
{\displaystyle \nu }
degrees of freedom , then according to the first definition,
1
/
X
{\displaystyle 1/X}
ν
{\displaystyle \nu }
ν
/
X
{\displaystyle \nu /X}
ν
{\displaystyle \nu }
Inverse-chi-square
Probability density function
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Cumulative distribution function
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ν
>
0
{\displaystyle \nu >0\,}
Support
x
∈
(
0
,
∞
)
{\displaystyle x\in (0,\infty )}
PDF
2
−
ν
/
2
Γ
(
ν
/
2
)
x
−
ν
/
2
−
1
e
−
1
/
(
2
x
)
{\displaystyle {\frac {2^{-\nu /2}}{\Gamma (\nu /2)}}\,x^{-\nu /2-1}e^{-1/(2x)}}
CDF
Γ
(
ν
2
,
1
2
x
)
/
Γ
(
ν
2
)
{\displaystyle \Gamma \left({\frac {\nu }{2}},{\frac {1}{2x}}\right)/\Gamma \left({\frac {\nu }{2}}\right)}
Mean
1
ν
−
2
{\displaystyle {\frac {1}{\nu -2}}}
ν
>
2
{\displaystyle \nu >2\,}
Mode
1
ν
+
2
{\displaystyle {\frac {1}{\nu +2}}}
Variance
2
(
ν
−
2
)
2
(
ν
−
4
)
{\displaystyle {\frac {2}{(\nu -2)^{2}(\nu -4)}}}
ν
>
4
{\displaystyle \nu >4\,}
Skewness
4
ν
−
6
2
(
ν
−
4
)
{\displaystyle {\frac {4}{\nu -6}}{\sqrt {2(\nu -4)}}}
ν
>
6
{\displaystyle \nu >6\,}
Excess kurtosis
12
(
5
ν
−
22
)
(
ν
−
6
)
(
ν
−
8
)
{\displaystyle {\frac {12(5\nu -22)}{(\nu -6)(\nu -8)}}}
ν
>
8
{\displaystyle \nu >8\,}
Entropy
ν
2
+
ln
(
1
2
Γ
(
ν
2
)
)
{\displaystyle {\frac {\nu }{2}}\!+\!\ln \left({\frac {1}{2}}\Gamma \left({\frac {\nu }{2}}\right)\right)}
−
(
1
+
ν
2
)
ψ
(
ν
2
)
{\displaystyle \!-\!\left(1\!+\!{\frac {\nu }{2}}\right)\psi \left({\frac {\nu }{2}}\right)}
MGF
2
Γ
(
ν
2
)
(
−
t
2
i
)
ν
4
K
ν
2
(
−
2
t
)
{\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-t}{2i}}\right)^{\!\!{\frac {\nu }{4}}}\!\!K_{\frac {\nu }{2}}\left({\sqrt {-2t}}\right)}
CF
2
Γ
(
ν
2
)
(
−
i
t
2
)
ν
4
K
ν
2
(
−
2
i
t
)
{\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-it}{2}}\right)^{\!\!{\frac {\nu }{4}}}\!\!K_{\frac {\nu }{2}}\left({\sqrt {-2it}}\right)}
This distribution arises in Bayesian statistics (spam filtering in particular).
It is a continuous distribution with a probability density function . The first definition yields a density function
f
(
x
;
ν
)
=
2
−
ν
/
2
Γ
(
ν
/
2
)
x
−
ν
/
2
−
1
e
−
1
/
(
2
x
)
{\displaystyle f(x;\nu )={\frac {2^{-\nu /2}}{\Gamma (\nu /2)}}\,x^{-\nu /2-1}e^{-1/(2x)}}
The second definition yields a density function
f
(
x
;
ν
)
=
(
ν
/
2
)
ν
/
2
Γ
(
ν
/
2
)
x
−
ν
/
2
−
1
e
−
ν
/
(
2
x
)
{\displaystyle f(x;\nu )={\frac {(\nu /2)^{\nu /2}}{\Gamma (\nu /2)}}x^{-\nu /2-1}e^{-\nu /(2x)}}
In both cases,
x
>
0
{\displaystyle x>0}
ν
{\displaystyle \nu }
degrees of freedom parameter. This article will deal with the first definition only. Both definitions are special cases of the scale-inverse-chi-square distribution . For the first definition
σ
2
=
1
/
ν
{\displaystyle \sigma ^{2}=1/\nu }
σ
2
=
1
{\displaystyle \sigma ^{2}=1}
chi-square : If
X
∼
χ
2
(
ν
)
{\displaystyle X\sim \chi ^{2}(\nu )}
Y
=
1
X
{\displaystyle Y={\frac {1}{X}}}
Y
∼
Inv-
χ
2
(
ν
)
{\displaystyle Y~\sim {\mbox{Inv-}}\chi ^{2}(\nu )}
Inverse gamma with
α
=
ν
2
{\displaystyle \alpha ={\frac {\nu }{2}}}
β
=
1
2
{\displaystyle \beta ={\frac {1}{2}}}
See also 
