Non-central chi-squared distribution

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non-central -distribution

A continuous probability distribution concentrated on the positive semi-axis with density

where is the number of degrees of freedom and the parameter of non-centrality. For this density is that of the ordinary (central) "chi-squared" distribution. The characteristic function of a non-central "chi-squared" distribution is

the mathematical expectation and variance (cf. Dispersion) are and , respectively. A non-central "chi-squared" distribution belongs to the class of infinitely-divisible distributions (cf. Infinitely-divisible distribution).

As a rule, a non-central "chi-squared" distribution appears as the distribution of the sum of squares of independent random variables having normal distributions with non-zero means and unit variance; more precisely, the sum has a non-central "chi-squared" distribution with degrees of freedom and non-centrality parameter . The sum of several mutually independent random variables with a non-central "chi-squared" distribution has a distribution of the same type and its parameters are the sums of the corresponding parameters of the summands.

If is even, then the distribution function of a non-central "chi-squared" distribution is given by for and for by

This formula establishes a link between a non-central "chi-squared" distribution and a Poisson distribution. Namely, if and have Poisson distributions with parameters and , respectively, then for any positive integer ,

A non-central "chi-squared" distribution often arises in problems of mathematical statistics concerned with the study of the power of tests of "chi-squared" type. Since tables of non-central "chi-squared" distributions are fairly complete, various approximations by means of a "chi-squared" and a normal distribution are widely used in statistical applications.


[1] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)
[2] M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1979)
[3] P.B. Patnaik, "The non-central - and -distributions and their applications" Biometrica , 36 (1949) pp. 202–232



[a1] N.L. Johnson, S. Kotz, "Distributions in statistics" , 2. Continuous univariate distributions , Wiley (1970)
How to Cite This Entry:
Non-central chi-squared distribution. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article