Noncentral chi distribution

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