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 *χ*^{2}
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.

The *χ*^{2} distribution is actually a
simple special case of the noncentral chi-square distribution. One way to generate
random numbers with a *χ*^{2} 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.

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')

`ncx2cdf`

| `ncx2inv`

| `ncx2pdf`

| `ncx2rnd`

| `ncx2stat`

| `random`