Non-central chi-squared distribution
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.
References
[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 ![]() ![]() |
Comments
References
[a1] | N.L. Johnson, S. Kotz, "Distributions in statistics" , 2. Continuous univariate distributions , Wiley (1970) |
Non-central chi-squared distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-central_chi-squared_distribution&oldid=14624