Evaluates the complement of the chi-squared distribution.

Synopsis

#include <imsls.h>

float imsls_f_complementary_chi_squared_cdf (float chi_squared, float df)

The type double function is imsls_d_complementary_chi_squared_cdf.

Required Arguments

float chi_squared   (Input)
Argument for which  Pr(x  > chi_squared)  is to be evaluated.

float df   (Input)
Number of degrees of freedom of the chi-squared distribution. Argument df must be greater than 0.5.

Return Value

The probability that a chi-squared random variable takes a value greater than chi_squared.

Description

Function imsls_f_complementary_chi_squared_cdf evaluates the distribution function,  , of a chi-squared random variable x = chi_squared with ν = df. Then,

where Γ (×) is the gamma function. The value of the distribution function at the point x is the probability that the random variable takes a value greater than x.

For ν > 343, imsls_f_complementary_chi_squared_cdf uses the Wilson-Hilferty approximation (Abramowitz and Stegun 1964, Equation 26.4.17) to the normal distribution, and function imsls_f_normal_cdf is used to evaluate the normal distribution function.

For ν £ 343, imsls_f_complementary_chi_squared_cdf uses series expansions to evaluate the distribution function. If x < max (ν / 2, 26), imsls_f_complementary_chi_squared_cdf uses the series 6.5.29 in Abramowitz and Stegun (1964); otherwise, it uses the asymptotic expansion 6.5.32 in Abramowitz and Stegun.

This function provides higher right tail accuracy for the chi-squared distribution.

Figure 11-3  Plot of Fx (x, df)

Example

In this example, we find the probability that X, a chi-squared random variable, is less than 0.15 and the probability that X is greater than 3.0.

 

#include <imsls.h>

#include <stdio.h>

 

#define DOUBLE

 

void main()

{

       float chi_squared = 0.15;

       float df = 2.0;

       float p;

 

       p = imsls_f_chi_squared_cdf(chi_squared, df);

       printf(" The probability that chi-squared\n");

       printf(" with df = %1.0f is less than %4.2f is %6.4f\n",

              df, chi_squared, p);

 

       chi_squared = 3.0;

       p = imsls_f_complementary_chi_squared_cdf(chi_squared, df);

       printf(" The probability that chi-squared\n");

       printf(" with df = %1.0f is greater than %4.2f is %6.4f\n",

              df, chi_squared, p);

}

Output

The probability that chi-squared

 with df = 2 is less than 0.15 is 0.0723

The probability that chi-squared

 with df = 2 is greater than 3.00 is 0.2231

Informational Errors

IMSLS_COMP_CHISQ_ZERO                       Since “chi_squared” = # is less than zero, the distribution function is one at “chi_squared”..

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