Non-central chi-squared 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.
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Non-central chi-squared distribution. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Non-central_chi-squared_distribution&oldid=28548