Noncentral chi
Parameters

degrees of freedom

Support
PDF
CDF with Marcum Q-function
Mean
Variance , where is the mean

In probability theory and statistics, the noncentral chi distribution[1] is a noncentral generalization of the chi distribution. It is also known as the generalized Rayleigh distribution.

Definition

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

Probability density function

The probability density function (pdf) is

where is a modified Bessel function of the first kind.

Raw moments

The first few raw moments are:

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

Bivariate non-central chi distribution

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.[2][3] If either or both or the distribution is a noncentral bivariate chi distribution.

  • 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.

References

  1. J. H. Park (1961). "Moments of the Generalized Rayleigh Distribution". Quarterly of Applied Mathematics. 19 (1): 45–49. doi:10.1090/qam/119222. JSTOR 43634840.
  2. Marakatha Krishnan (1967). "The Noncentral Bivariate Chi Distribution". SIAM Review. 9 (4): 708–714. Bibcode:1967SIAMR...9..708K. doi:10.1137/1009111.
  3. P. R. Krishnaiah, P. Hagis, Jr. and L. Steinberg (1963). "A note on the bivariate chi distribution". SIAM Review. 5 (2): 140–144. Bibcode:1963SIAMR...5..140K. doi:10.1137/1005034. JSTOR 2027477.{{cite journal}}: CS1 maint: multiple names: authors list (link)
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.