Chapter 11: Probability Distribution Functions and Inverses

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 (float  x, float  df,  float lambda)

The type double function is imsls_d_non_central_chi_sq_pdf.

Required Arguments

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

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

float  lambda  (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 mi and variances s2i, 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 l 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 l ¤ 2. Thus, the probability density function is given by:

          

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

where G(.) is the gamma function.  The above representation of F(x, k, lcan 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, l = 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 l = 40.

#include <imsls.h>

#include <stdio.h>

int main()

{

   int i;

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

   float df = 100;

   float lambda =40.0;

   float 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

 


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