chi_squared_cdf

Evaluates the chi‑squared cumulative distribution function (CDF).

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 0.

Return Value

The probability p 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(x∣v) , of a chi-squared random variable x = chi_squared with ν = df degrees of freedom, where:

 

and 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 v > vmax = 1.e7, imsls_f_chi_squared_cdf uses the Wilson-Hilferty approximation (Abramowitz and Stegun [A&S] 1964, Equation 26.4.17) for p in terms of the normal CDF, which is evaluated using function imsls_f_normal_cdf.

For v ≤ vmax, imsls_f_chi_squared_cdf uses series expansions to evaluate p: for x < ν, imsls_f_chi_squared_cdf calculates p using A&S series 6.5.29, and for x > ν, imsls_f_chi_squared_cdf calculates p using the continued fraction expansion of the incomplete gamma function given in A&S equation 6.5.31.

 

Figure 1,  Plot of Fx (x, df)

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>

#include <stdio.h>

 

int main()

{

    float  chi_squared = 0.15, df = 2.0, p;

 

    p    = imsls_f_chi_squared_cdf(chi_squared, df);

    printf("The probability that chi-squared"

        " with %1.0f df is less than %4.2f is %5.4f\n",

        df, chi_squared, p);

 

    chi_squared = 3.0;

    p    = 1.0 - imsls_f_chi_squared_cdf(chi_squared, df);

    printf("The probability that chi-squared"

        " with %1.0f df is greater than %3.1f is %5.4f\n",

        df, chi_squared, 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.