Noncentral chi distribution

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Noncentral chi
Parameters

degrees of freedom

Support
PDF
CDF with Marcum Q-function
Mean
Variance

In probability theory and statistics, the noncentral chi distribution is a generalization of the chi distribution. If are k independent, normally distributed random variables with means and variances , then the statistic

is distributed according to the noncentral chi distribution. The noncentral chi distribution has two parameters: which specifies the number of degrees of freedom (i.e. the number of ), and which is related to the mean of the random variables by:

Properties[edit]

Probability density function[edit]

The probability density function (pdf) is

where is a modified Bessel function of the first kind.

Raw moments[edit]

The first few raw moments are:

where is the generalized Laguerre polynomial. Note that the 2th moment is the same as the th moment of the noncentral chi-squared distribution with being replaced by .

Differential equation[edit]

The pdf of the noncentral chi distribution is a solution to the following differential equation:

Bivariate non-central chi distribution[edit]

Let , be a set of n independent and identically distributed bivariate normal random vectors with marginal distributions , correlation , and mean vector and covariance matrix

with positive definite. Define

Then the joint distribution of U, V is central or noncentral bivariate chi distribution with n degrees of freedom.[1][2] If either or both or the distribution is a noncentral bivariate chi distribution.

Related distributions[edit]

  • If is a random variable with the non-central chi distribution, the random variable will have the noncentral chi-squared distribution. Other related distributions may be seen there.
  • If is chi distributed: then is also non-central chi distributed: . In other words, the chi distribution is a special case of the non-central chi distribution (i.e., with a non-centrality parameter of zero).
  • A noncentral chi distribution with 2 degrees of freedom is equivalent to a Rice distribution with .
  • If X follows a noncentral chi distribution with 1 degree of freedom and noncentrality parameter λ, then σX follows a folded normal distribution whose parameters are equal to σλ and σ2 for any value of σ.

Applications[edit]

The Euclidean norm of a multivariate normally distributed random vector follows a noncentral chi distribution.

References[edit]

  1. ^ Marakatha Krishnan (1967). "The Noncentral Bivariate Chi Distribution". SIAM Review. 9 (4): 708–714. doi:10.1137/1009111. 
  2. ^ P. R. Krishnaiah, P. Hagis, Jr. and L. Steinberg (1963). "A note on the bivariate chi distribution". SIAM Review. 5: 140–144. doi:10.1137/1005034. JSTOR 2027477.