non_central_chi_sq_pdf

Evaluates the noncentral chi-squared probability density function.

Synopsis

#include <imsls.h>

float imsls_f_non_central_chi_sq_pdf (floatx, floatdf, floatlambda)

The type doublefunction is imsls_d_non_central_chi_sq_pdf.

Required Arguments

floatx (Input)
Argument for which the noncentral chi-squared probability density function is to be evaluated. x must be greater than or equal to 0.

floatdf (Input)
Number of degrees of freedom of the noncentral chi-squared distribution. df must be greater than 0.

floatlambda (Input)
Noncentrality parameter. lambda must be greater than or equal to 0.

Return Value

The probability density associated with a noncentral chi-squared random variable with value x.

Description

The noncentral chi‑squared distribution is a generalization of the chi-squared distribution. If {Xi}are k independent, normally distributed random variables with means μi and variances σ2i, then the random variable:

 

is distributed according to the noncentral chi-squared distribution. The noncentral chi-squared distribution has two parameters: k which specifies the number of degrees of freedom (i.e. the number of Xi), and λ which is related to the mean of the random variables Xi by:

 

The noncentral chi-squared distribution is equivalent to a (central) chi-squared distribution with k + 2i degrees of freedom, where i is the value of a Poisson distributed random variable with parameter λ / 2. Thus, the probability density function is given by:

 

where the (central) chi-squared PDF f(x∣k)is given by:

 

where Γ (⋅)is the gamma function. The above representation of F(x∣k,λ)can be shown to be equivalent to the representation:

 

 

Function imsls_f_non_central_chi_sq_pdf evaluates the probability density function of a noncentral chi-squared random variable with df degrees of freedom and noncentrality parameter lambda, corresponding to k = df, λ = lambda, and x = x.

Function imsls_f_non_central_chi_sq evaluates the cumulative distribution function incorporating the above probability density function.

With a noncentrality parameter of zero, the noncentral chi-squared distribution is the same as the central chi-squared distribution.

Example

This example calculates the noncentral chi-squared distribution for a distribution with 100 degrees of freedom and noncentrality parameter λ = 40.

 

#include <imsls.h>

#include <stdio.h>

 

int main()

{

    int i;

    float x[] = {0, 8, 40, 136, 280, 400};

    float df = 100, lambda = 40.0, pdfv;

 

    printf ("\n\n df: %4.0f;  lambda: %4.0f\n\n",

        df, lambda);

    printf ("    x       pdf(x)\n");

 

    for (i=0; i<6; i++) {

        pdfv = imsls_f_non_central_chi_sq_pdf(x[i], df, lambda);

        printf (" %5.0f  %12.4e\n",x[i], pdfv);

    }

}

 

Output

 

df: 100; lambda: 40

 

x pdf(x)

0 0.0000e+000

8 4.7644e-044

40 3.4621e-014

136 2.1092e-002

280 4.0027e-010

400 1.1250e-022