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

Definition

are k independent, normally distributed, then the statistic

which specifies the number of degrees of freedom by:

Properties

Probability density function

The probability density function (pdf) is

is a modified Bessel function of the first kind.

Raw moments

The first few raw moments are:

is a Laguerre functionth moment of the noncentral chi-squared distribution.

Bivariate non-central chi distribution

, be a set of n independent and identically distributed bivariate normal, and mean vector and covariance matrix

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.

Related distributions

• will have the noncentral chi-squared distribution. Other related distributions may be seen there.

• is chi. 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.

• 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