degrees of freedom 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.

are k independent, normally distributed random variables with means

σ

, 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

which is related to the mean of the random variables

by: The probability density function (pdf) is where

is a modified Bessel function of the first kind.

The first few raw moments are: where

is a Laguerre function.

th moment of the noncentral chi-squared distribution with

, be a set of n independent and identically distributed bivariate normal random vectors with marginal distributions

, 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.

the distribution is a noncentral bivariate chi distribution.