g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_prob_students_t (g01ebc)

## 1  Purpose

nag_prob_students_t (g01ebc) returns the lower tail, upper tail or two tail probability for the Student's $t$-distribution with real degrees of freedom.

## 2  Specification

 #include #include
 double nag_prob_students_t (Nag_TailProbability tail, double t, double df, NagError *fail)

## 3  Description

The lower tail probability for the Student's $t$-distribution with $\nu$ degrees of freedom, $P\left(T\le t:\nu \right)$ is defined by:
 $P T≤t:ν = Γ ν+1 / 2 πν Γν/2 ∫ -∞ t 1+ T2ν -ν+1 / 2 dT , ν≥1 .$
Computationally, there are two situations:
(i) when $\nu <20$, a transformation of the beta distribution, ${P}_{\beta }\left(B\le \beta :a,b\right)$ is used
 $P T≤t:ν = 12 Pβ B≤ ν ν+t2 : ν/2, 12 when ​ t<0.0$
or
 $P T≤t:ν = 12 + 12 Pβ B≥ ν ν+t2 : ν/2, 12 when ​ t>0.0 ;$
(ii) when $\nu \ge 20$, an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see Hill (1970).

## 4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Hill G W (1970) Student's $t$-distribution Comm. ACM 13(10) 617–619

## 5  Arguments

1:     tailNag_TailProbabilityInput
On entry: indicates which tail the returned probability should represent.
${\mathbf{tail}}=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., $P\left(T\ge t:\nu \right)$.
${\mathbf{tail}}=\mathrm{Nag_TwoTailSignif}$
The two tail (significance level) probability is returned, i.e., $P\left(T\ge \left|t\right|:\nu \right)+P\left(T\le -\left|t\right|:\nu \right)$.
${\mathbf{tail}}=\mathrm{Nag_TwoTailConfid}$
The two tail (confidence interval) probability is returned, i.e., $P\left(T\le \left|t\right|:\nu \right)-P\left(T\le -\left|t\right|:\nu \right)$.
${\mathbf{tail}}=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., $P\left(T\le t:\nu \right)$.
Constraint: ${\mathbf{tail}}=\mathrm{Nag_UpperTail}$, $\mathrm{Nag_TwoTailSignif}$, $\mathrm{Nag_TwoTailConfid}$ or $\mathrm{Nag_LowerTail}$.
2:     tdoubleInput
On entry: $t$, the value of the Student's $t$ variate.
3:     dfdoubleInput
On entry: $\nu$, the degrees of freedom of the Student's $t$-distribution.
Constraint: ${\mathbf{df}}\ge 1.0$.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL_ARG_LT
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}\ge 1.0$.

## 7  Accuracy

The computed probability should be accurate to five significant places for reasonable probabilities but there will be some loss of accuracy for very low probabilities (less than ${10}^{-10}$), see Hastings and Peacock (1975).

The probabilities could also be obtained by using the appropriate transformation to a beta distribution (see Abramowitz and Stegun (1972)) and using nag_prob_beta_dist (g01eec). This function allows you to set the required accuracy.

## 9  Example

This example reads values from, and degrees of freedom for Student's $t$-distributions along with the required tail. The probabilities are calculated and printed until the end of data is reached.

### 9.1  Program Text

Program Text (g01ebce.c)

### 9.2  Program Data

Program Data (g01ebce.d)

### 9.3  Program Results

Program Results (g01ebce.r)