TOPICS
Search

Chi Distribution


The chi distribution with n degrees of freedom is the distribution followed by the square root of a chi-squared random variable. For n=1, the chi distribution is a half-normal distribution with theta=sqrt(pi/2). For n=2, it is a Rayleigh distribution with sigma=1. The chi distribution is implemented in the Wolfram Language as ChiDistribution[n].

ChiDistribution
ChiDistributionPlots

The probability density function and distribution function for this distribution are

P_n(x)=(2^(1-n/2)x^(n-1)e^(-x^2/2))/(Gamma(1/2n))
(1)
D_n(x)=P(1/2n,1/2x^2).
(2)

where P(a,z) is a regularized gamma function.

The rth raw moment is

 mu_r^'=(2^(r/2)Gamma(1/2(n+r)))/(Gamma(1/2n))
(3)

(Johnson et al. 1994, p. 421; Evans et al. 2000, p. 57; typo corrected), giving the first few as

mu_1^'=(sqrt(2)Gamma(1/2(n+1)))/(Gamma(1/2n))
(4)
mu_2^'=n
(5)
mu_3^'=(2sqrt(2)Gamma(1/2(n+3)))/(Gamma(1/2n))
(6)
mu_4^'=n(n+2).
(7)

The mean, variance, skewness, and kurtosis excess are given by

mu=(sqrt(2)Gamma(1/2(n+1)))/(Gamma(1/2n))
(8)
sigma^2=(2[Gamma(1/2n)Gamma(1+1/2n)-Gamma^2(1/2(n+1))])/(Gamma^2(1/2n))
(9)
gamma_1=(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))
(10)
gamma_2=(-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).
(11)

See also

Chi-Squared Distribution, Half-Normal Distribution, Rayleigh Distribution

Explore with Wolfram|Alpha

References

Evans, M.; Hastings, N.; and Peacock, B. "Chi Distribution." §8.3 in Statistical Distributions, 3rd ed. New York: Wiley, p. 57, 2000.Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous Univariate Distributions, Vol. 1, 2nd ed. Boston, MA: Houghton Mifflin, 1994.

Referenced on Wolfram|Alpha

Chi Distribution

Cite this as:

Weisstein, Eric W. "Chi Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChiDistribution.html

Subject classifications