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 ofχ²probabilities 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

$F (x | ν, δ) = \sum_{j = 0}^{\infty} (\frac{{(\frac{1}{2} δ)}^{j}}{j!} e^{\frac{- δ}{2}}) \Pr [χ_{ν + 2 j}^{2} \leq x]$

where δ is the noncentrality parameter.

Background

Theχ²distribution is actually a simple special case of the noncentral chi-square distribution. One way to generate random numbers with aχ²distribution (withνdegrees of freedom) is to sum the squares ofνstandard 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 theRician Distributionor generalizedRayleigh Distribution.

Examples

Compute Noncentral Chi-Square Distribution pdf

Open Live Script

Compute the pdf of a noncentral chi-square distribution with degrees of freedomV = 4and noncentrality parameterDELTA = 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) holdonplot(x,chi2,'g--','LineWidth',2) legend('ncx2','chi2')

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent ncx2, chi2.