Noncentral Chi-Squared Distribution

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

			(1)
			(2)
			(3)

where  is a modified Bessel function of the first kind and  is a regularized confluent hypergeometric limit function. It is implemented in the Wolfram Language as NoncentralChiSquareDistribution[r, lambda].

The mean, variance, skewness, and kurtosis excess are

			(4)
			(5)
			(6)
			(7)

The raw moments can be calculated analytically as

(8)

The first few are therefore

			(9)
			(10)
			(11)

The first few central moments are

			(12)
			(13)
			(14)

REFERENCES:

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.