Noncentral chi distribution

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 $$\mu_i$$ 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 $$\lambda$$ 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 $$I_\nu(z)$$ 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-square distribution with $$\lambda$$ being replaced by $$\lambda^2$$.

Related distributions

 * If $$X$$ is a random variable with the non-central chi distribution, the random variable $$X^2$$ will have the noncentral chi-square distribution. Other related distributions may be seen there.


 * If $$X$$ is chi distributed: $$X \sim \chi_k$$ then $$X$$ is also non-central chi distributed: $$X \sim NC\chi_k(0)$$. 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 on 2 degrees of freedom is equivalent to a Rice distribution with $$\sigma=1$$.