Noncentral chi distribution

Noncentral chi

Parameters	$k > 0\,$ degrees of freedom $\lambda > 0\,$
Support	$x \in [0; +\infty)\,$
PDF	$\frac{e^{-(x^2+\lambda^2)/2}x^k\lambda} {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)$
Mean	$\sqrt{\frac{\pi}{2}}L_{1/2}^{(k/2-1)}\left(\frac{-\lambda^2}{2}\right)\,$
Variance	$k+\lambda^2-\mu^2\,$

In probability theory and statistics, the noncentral chi distribution is a generalization of the chi distribution. If X_i are k independent, normally distributed random variables with means μ_i and variances $\sigma_i^2$, then the statistic

$Z = \sqrt{\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}$

is distributed according to the noncentral chi distribution. The noncentral chi distribution has two parameters: k which specifies the number of degrees of freedom (i.e. the number of X_i), and λ which is related to the mean of the random variables X_i by:

$\lambda=\sqrt{\sum_1^k \left(\frac{\mu_i}{\sigma_i}\right)^2}$

Properties

The probability density function is

$f(x;k,\lambda)=\frac{e^{-(x^2+\lambda^2)/2}x^k\lambda} {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)$

where I_ν(z) is a modified Bessel function of the first kind.

The first few raw moments are:

$\mu^'_1=\sqrt{\frac{\pi}{2}}L_{1/2}^{(k/2-1)}\left(\frac{-\lambda^2}{2}\right)$

$\mu^'_2=k+\lambda^2$

$\mu^'_3=3\sqrt{\frac{\pi}{2}}L_{3/2}^{(k/2-1)}\left(\frac{-\lambda^2}{2}\right)$

$\mu^'_4=(k+\lambda^2)^2+2(k+2\lambda^2)$

where $L_n^{(a)}(z)$ is the generalized Laguerre polynomial. Note that the 2nth moment is the same as the nth moment of the noncentral chi-squared distribution with λ being replaced by λ².

Related distributions

• If X is a random variable with the non-central chi distribution, the random variable X² will have the noncentral chi-squared distribution. Other related distributions may be seen there.

• If X is chi distributed: X∼χ_k then X is also non-central chi distributed: X∼NCχ_k(0). 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 σ = 1.

• 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.