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Noncentral Chi-Squared Distribution

The noncentral chi-squared distribution with noncentrality parameter lambda is given by

P_r(x)=(e^(-(x+lambda)/2)x^(r/2-1))/(2^(r/2))sum_(k=0)^(infty)((lambdax)^k)/(2^(2k)k!Gamma(k+1/2r))
(1)
=(e^(-(x+lambda)/2)x^((r-1)/2)sqrt(lambda))/(2(lambdax)^(r/4))I_(r/2-1)(sqrt(lambdax))
(2)
=2^(-r/2)e^(-(lambda+x)/2)x^(r/2-1)_0F^~_1(;1/2r;1/4lambdax),
(3)

where I_n(x) is a modified Bessel function of the first kind and _0F^~_1 is a regularized confluent hypergeometric limit function. It is implemented in the Wolfram Language as NoncentralChiSquareDistribution[r, lambda].

For x>=0, its distribution function and survival function can be written in terms of the generalized Marcum Q-function as

D_r(x)=1-Q_(r/2)(sqrt(lambda),sqrt(x))
(4)
1-D_r(x)=Q_(r/2)(sqrt(lambda),sqrt(x)).
(5)

(Peterson et al. 1995, pp. 672-681).

The mean, variance, skewness, and kurtosis excess are

mu=lambda+r
(6)
sigma^2=2(2lambda+r)
(7)
gamma_1=(2sqrt(2)(3lambda+r))/((2lambda+r)^(3/2))
(8)
gamma_2=(12(4lambda+r))/((2lambda+r)^2).
(9)

The raw moments can be calculated analytically as

mu_n^'=2^ne^(-lambda/2)Gamma(n+1/2r)_1F^~_1(n+1/2r,1/2r,1/2lambda).
(10)

The first few are therefore

mu_1^'=r+lambda
(11)
mu_2^'=r^2+2r(lambda+1)+lambda(lambda+4)
(12)
mu_3^'=r^3+3r^2(lambda+2)+lambda(lambda^2+12lambda+24)+r(3lambda^2+18lambda+8).
(13)

The first few central moments are

mu_2=2(r+2lambda)
(14)
mu_3=8(r+3lambda)
(15)
mu_4=12[r^2+4r(lambda+1)+4lambda(lambda+4)].
(16)

See also

Chi-Squared Distribution, Marcum Q-Function

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References

Peterson, R. L.; Ziemer, R. E.; and Borth, D. E. "Numerical Approximations for the Chi-Squared Probability Distribution and Marcum's Q-Function." Appendix F in Introduction to Spread-Spectrum Communications. Englewood Cliffs, NJ: Prentice-Hall, pp. 672-681, 1995.Stuart, A.; and Ord, J. K. Kendall's Advanced Theory of Statistics, Vol. 2A: Classical Inference & the Linear Model, 6th ed. New York: Oxford University Press, p. 865, 1999.

Referenced on Wolfram|Alpha

Noncentral Chi-Squared Distribution

Cite this as:

Weisstein, Eric W. "Noncentral Chi-Squared Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html

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