Noncentral chi distribution


In probability theory and statistics, the noncentral chi distribution is a generalization of the chi 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, and which is related to the mean of the random variables by:

Properties

Probability density function

The probability density function 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.
If either or both or the distribution is a noncentral bivariate chi distribution.

Related distributions