tCdf

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

Synopsis

tCdf (t, df)

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 tCdf evaluates the cumulative distribution function of a Student’s t random variable with ν = df degrees of freedom. If $t^2\geq\nu$, the following identity relating the Student’s t cumulative distribution function, $F(t,\nu)$ to the incomplete beta ratio function $I_x (a,b)$ is used:

$$F(t|\nu) = \frac{1}{2} I_x \left(\frac{\nu}{2},\frac{1}{2}\right), t \leq 0, t^2 \geq \nu$$

where

$$x = \frac{v}{t^2+v}$$

and

$$F(t|\nu) = 1 - F(-t,\nu), t> 0, t^2 \geq \nu$$

If $t^2<\nu$, the solution space is partitioned into four algorithms as follows: If $\nu\geq 64$ and $t^2/\nu\leq 0.1$, a Cornish-Fisher expansion is used to evaluate the distribution function. If $\nu<64$ 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 $\nu<64$ and an integer and $|t|\geq 2.0$, a series given by Hill (1970) that converges well for large values of t is used. For the remaining $t^2<\nu$ cases, $F(t|\nu)$ is calculated using the identity:

$$F(t|\nu) = I_x \left(\frac{\nu}{2},\frac{\nu}{2}\right)$$

where

$$x = \frac{t + \sqrt{t^2 + \nu}}{2 \sqrt{t^2 + \nu}}$$

Figure 11.9 — Plot of $F_t(t,6.0)$

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.

from __future__ import print_function
from numpy import *
from pyimsl.stat.tCdf import tCdf

t = 2.447
df = 6.0
pr = 2 * tCdf(-t, df)
print("Pr(|t(6)| > 2.447) = %6.4f\n" % pr)

Output

Pr(|t(6)| > 2.447) = 0.0500