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
where δ is the noncentrality parameter.
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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 or generalized Rayleigh distribution.
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Example
The following commands generate a plot of the noncentral chi-square
pdf.
x = (0:0.1:10)';
p1 = ncx2pdf(x,4,2);
p = chi2pdf(x,4);
plot(x,p,'-',x,p1,'-')
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