Noncentral Chi-Square Distribution

Definition

There are many equivalent formulas for the noncentral chi-square distribution function. One formulation uses a modified Bessel function of the first kind. Another uses the generalized Laguerre polynomials. The cumulative distribution function is computed using a weighted sum of χ² probabilities with the weights equal to the probabilities of a Poisson distribution. The Poisson parameter is one-half of the noncentrality parameter of the noncentral chi-square

$$F(x|\nu ,\delta )={\displaystyle \sum _{j=0}^{\infty }\left(\frac{{\left(\frac{1}{2}\delta \right)}^j}{j!}{e}^{\frac{-\delta }{2}}\right)}\mathrm{Pr}\left[{\chi }_{\nu +2j}^2\le x\right]$$

where δ is the noncentrality parameter.

Background

The χ² distribution is actually a simple special case of the noncentral chi-square distribution. One way to generate random numbers with a χ² distribution (with ν degrees of freedom) is to sum the squares of ν standard normal random numbers (mean equal to zero.)

What if the normally distributed quantities have a mean other than zero? The sum of squares of these numbers yields the noncentral chi-square distribution. The noncentral chi-square distribution requires two parameters: the degrees of freedom and the noncentrality parameter. The noncentrality parameter is the sum of the squared means of the normally distributed quantities.

The noncentral chi-square has scientific application in thermodynamics and signal processing. The literature in these areas may refer to it as the Rician Distribution or generalized Rayleigh Distribution.

Examples

Compute Noncentral Chi-Square Distribution pdf

Compute the pdf of a noncentral chi-square distribution with degrees of freedom V = 4 and noncentrality parameter DELTA = 2. For comparison, also compute the pdf of a chi-square distribution with the same degrees of freedom.

x = (0:0.1:10)';
ncx2 = ncx2pdf(x,4,2);
chi2 = chi2pdf(x,4);

Plot the pdf of the noncentral chi-square distribution on the same figure as the pdf of the chi-square distribution.

figure;
plot(x,ncx2,'b-','LineWidth',2)
hold on
plot(x,chi2,'g--','LineWidth',2)
legend('ncx2','chi2')