Evaluates the chi-squared distribution function.
Synopsis
#include <imsls.h>
float imsls_f_chi_squared_cdf (float chi_squared, float df)
The type double function is imsls_d_chi_squared_cdf.
Required Arguments
float
chi_squared (Input)
Argument for which the chi-squared
distribution function is to be evaluated.
float df
(Input)
Number of degrees of freedom of the chi-squared distribution.
Argument df must
be greater than or equal to 0.5.
Return Value
The probability that a chi-squared random variable takes a value less than or equal to chi_squared.
Description
Function imsls_f_chi_squared_cdf evaluates the distribution function, F, 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 less than or equal to x.
For ν > 65, imsls_f_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 £ 65, imsls_f_chi_squared_cdf uses series expansions to evaluate the distribution function. If x < max (ν / 2, 26), imsls_f_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.
Example
Suppose X is a chi-squared random variable with two
degrees of freedom. In this example, we find the probability that X is
less than 0.15 and the probability that
X is greater than 3.0.
#include <imsls.h>
int main()
{
float chi_squared = 0.15;
float df = 2.0;
float p;
p = imsls_f_chi_squared_cdf(chi_squared, df);
printf("%s %s %6.4f\n", "The probability that chi-squared\n",
"with 2 df is less than 0.15 is", p);
chi_squared = 3.0;
p = 1.0 - imsls_f_chi_squared_cdf(chi_squared, df);
printf("%s %s %6.4f\n", "The probability that chi-squared\n",
"with 2 df is greater than 3.0 is", p);
}
Output
The probability that chi-squared
with 2 df is less than 0.15 is 0.0723
The probability that chi-squared
with 2 df is greater than 3.0 is 0.2231
Informational Errors
IMSLS_ARG_LESS_THAN_ZERO Since “chi_squared” = # is less than zero, the distribution function is zero at “chi_squared.”
Alert Errors
IMSLS_NORMAL_UNDERFLOW Using the normal distribution for large degrees of freedom, underflow would have occurred.
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