t_cdf

Evaluates the Student’s t cumulative distribution function (CDF).

Synopsis

#include <imsl.h>

float imsl_f_t_cdf (float t, float df)

The type double function is imsl_d_t_cdf.

Required Arguments

float t (Input)
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.

Return Value

The probability that a Student’s t random variable takes a value less than or equal to the input t.

Description

Function imsl_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 TCDF(t, ν) to the incomplete beta ratio function Ix (a, b) 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, TCDF(t, ν) is calculated using the identity:

 

where

 

 

Figure 1,  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 <imsl.h>

#include <stdio.h>

 

int main ()

{

    float  t = 2.447, df = 6.0, p;

 

    p  = 2.0*imsl_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