The chi distribution with 
degrees of freedom is the distribution followed by the square
root of a chi-squared random variable. For 
, the 
distribution is a half-normal
distribution with 
.
For 
,
it is a Rayleigh distribution with 
. The chi distribution is implemented
in the Wolfram Language as ChiDistribution[n].
The probability density function and distribution function for this distribution are
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(1)
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(2)
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where 
is a regularized gamma function.
The 
th
raw moment is
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(3)
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(Johnson et al. 1994, p. 421; Evans et al. 2000, p. 57; typo corrected), giving the first few as
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(4)
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(5)
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(6)
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(7)
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The mean, variance, skewness, and kurtosis excess are given by
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(8)
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 |  | ![(2[Gamma(1/2n)Gamma(1+1/2n)-Gamma^2(1/2(n+1))])/(Gamma^2(1/2n))](/images/equations/ChiDistribution/Inline32.svg) |
(9)
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 |  | ![(2Gamma^3(1/2(n+1))-3Gamma(1/2n)Gamma(1/2(n+1))Gamma(1+1/2n))/([Gamma(1/2n)Gamma(1+1/2n)-Gamma^2(1/2(n+1))]^(3/2))+(Gamma^2(1/2n)Gamma((3+n)/2))/([Gamma(1/2n)Gamma(1+1/2n)-Gamma^2(1/2(n+1))]^(3/2))](/images/equations/ChiDistribution/Inline35.svg) |
(10)
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 |  | ![(-3Gamma^4(1/2(n+1))+6Gamma(1/2n)Gamma^2(1/2(n+1))Gamma(1+1/2n))/([Gamma(1/2n)Gamma((2+n)/2)-Gamma^2(1/2(n+1))]^2)+(-4Gamma^2(1/2n)Gamma(1/2(n+1))Gamma((3+n)/2)+Gamma^3(1/2n)Gamma((4+n)/2))/([Gamma(1/2n)Gamma((2+n)/2)-Gamma^2(1/2(n+1))]^2).](/images/equations/ChiDistribution/Inline38.svg) |
(11)
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