This function evaluates the Student’s t cumulative distribution function.

Function TDF evaluates the cumulative distribution function of a Student’s t random variable with DF degrees of freedom. If the square of T is greater than or equal to DF, the relationship of a t to an F random variable (and subsequently, to a beta random variable) is exploited, and routine BETDF is used. Otherwise, the method described by Hill (1970) is used. Let ν = DF. If ν is not an integer, if ν is greater than 19, or if ν is greater than 200, a Cornish‑Fisher expansion is used to evaluate the distribution function. If ν is less than 20 and ABS(T) is less than 2.0, a trigonometric series (see Abramowitz and Stegun 1964, equations 26.7.3 and 26.7.4, with some rearrangement) is used. For the remaining cases, a series given by Hill (1970) that converges well for large values of T is used.

If COMPLEMENT = .TRUE., the value of TDF at the point x is 1 ‑ p, where 1 ‑ p is the probability that the random variable takes a value greater than x. In those situations where the desired end result is 1 ‑ p, the user can achieve greater accuracy in the right tail region by using the result returned by TDF with the optional argument COMPLEMENT set to .TRUE. rather than by using 1 ‑ p where p is the result returned by TDF with COMPLEMENT set to .FALSE..

In this example, we find the probability that a t random variable with 6 degrees of freedom is greater in absolute value than 2.447. We use the fact that t is symmetric about 0.