Noncentral Chi-Square Distribution

R2026b

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

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

where δ is the noncentrality parameter.

Background

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.

Parameters

The noncentral chi-square distribution uses the following parameters.

Parameter	Description	Support
ν	Degrees of freedom	ν `= 1, 2, 3,...`
δ	Noncentrality parameter	δ ≥ `0`

Examples

Compute and Plot Noncentral Chi-Square Probability Density Function

Open Live Script

Compute the probability density function (pdf) of a noncentral chi-square distribution with 4 degrees of freedom and the noncentrality parameter delta=2. For comparison, also compute the pdf of a chi-square distribution with the same degrees of freedom.

x = 0:0.1:15;
nu = 4;
delta = 2;
ncx2 = ncx2pdf(x,nu,delta);
chi2 = chi2pdf(x,nu);

Plot the noncentral chi-square pdf and the chi-square pdf on the same figure.

figure
plot(x,ncx2,"b-",LineWidth=2)
hold on
grid on
plot(x,chi2,"r--",Linewidth=2)
xlabel("x")
ylabel("p")
legend("Noncentral chi-square pdf","Chi-square pdf")
hold off

Figure contains an axes object. The axes object with xlabel x, ylabel p contains 2 objects of type line. These objects represent Noncentral chi-square pdf, Chi-square pdf.