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

The type double function is imsl_d_t_cdf.

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

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

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.