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')

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