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
 |  |  | (1) |
 |  |  | (2) |
where 
is a regularized gamma function.
The 
th raw moment is
 | (3) |
(Johnson et al. 1994, p. 421; Evans et al. 2000, p. 57; typo corrected), giving the first few as
 |  |  | (4) |
 |  |  | (5) |
 |  |  | (6) |
 |  |  | (7) |
The mean, variance, skewness, and kurtosis excess are given by
 |  |  | (8) |
 |  | ![(2[Gamma(1/2n)Gamma(1+1/2n)-Gamma^2(1/2(n+1))])/(Gamma^2(1/2n))](http://mathworld.wolfram.com/images/equations/ChiDistribution/Inline32.svg) | (9) |
 |  | ![(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))](http://mathworld.wolfram.com/images/equations/ChiDistribution/Inline35.svg) | (10) |
 |  | ![(-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).](http://mathworld.wolfram.com/images/equations/ChiDistribution/Inline38.svg) | (11) |
