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)
| |||
(9)
| |||
(10)
| |||
(11)
|