t_cdf
Evaluates the Student’s t cumulative distribution function (CDF).
Synopsis
#include <imsls.h>
float imsls_f_t_cdf (float t, float df)
The type double function is imsls_d_t_cdf.
Required Arguments
float t (Input)
Argument for which the Student’s t cumulative distribution function is to be evaluated.
Argument for which the Student’s t cumulative distribution function is to be evaluated.
float df (Input)
Degrees of freedom. Argument df must be greater than or equal to 1.0.
Degrees of freedom. Argument df must be greater than or equal to 1.0.
Return Value
The probability that a Student’s t random variable takes a value less than or equal to the input t.
Description
Function imsls_f_t_cdf evaluates the cumulative distribution function of a Student’s t random variable with ν = df degrees of freedom. If t2 ≥ ν, the following identity relating the Student’s t cumulative distribution function, F(t, ν) to the incomplete beta ratio function is used:
where
and
If t2 < ν, the solution space is partitioned into four algorithms as follows: If ν ≥ 64 and t2/ν ≤ 0.1, a Cornish-Fisher expansion is used to evaluate the distribution function. If ν < 64 and an integer and |t| < 2.0, a trigonometric series is used (see Abramowitz and Stegun 1964, Equations 26.7.3 and 26.7.4 with some rearrangement). If ν < 64 and an integer and |t| ≥ 2.0, a series given by Hill (1970) that converges well for large values of t is used. For the remaining t2 < ν cases, F(t|ν) is calculated using the identity:
where
Figure 19, Plot of Ft (t, 6.0)
Example
This example finds the probability that a t random variable with 6 degrees of freedom is greater in absolute value than 2.447. The fact that t is symmetric about 0 is used.
#include <imsls.h>
#include <stdio.h>
int main ()
{
float t = 2.447, df = 6.0, p;
p = 2.0*imsls_f_t_cdf(-t,df);
printf("Pr(|t(%1.0f)| > %5.3f) = %6.4f\n", df, t, p);
}
Output
Pr(|t(6)| > 2.447) = 0.0500
