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:
Then the joint distribution of U, V is central or noncentral bivariate chi distribution with ndegrees of freedom.
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.
^Marakatha Krishnan (1967). "The Noncentral Bivariate Chi Distribution". SIAM Review. 9 (4): 708–714. doi:10.1137/1009111.
^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. JSTOR2027477.CS1 maint: multiple names: authors list (link)