g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rand_students_t (g05snc)

## 1  Purpose

nag_rand_students_t (g05snc) generates a vector of pseudorandom numbers taken from a Student's $t$-distribution with $\nu$ degrees of freedom.

## 2  Specification

 #include #include
 void nag_rand_students_t (Integer n, Integer df, Integer state[], double x[], NagError *fail)

## 3  Description

The distribution has PDF (probability density function)
 $fx= ν-12 ! 12ν-1!πν 1+x2ν 12ν+1 .$
nag_rand_students_t (g05snc) calculates the values
 $yiνzi, i= 1,…,n$
where the ${y}_{i}$ are generated by nag_rand_normal (g05skc) with mean $0$ and variance $1.0$, and the ${z}_{i}$ are generated by nag_rand_gamma (g05sjc) with parameters $\frac{1}{2}\nu$ and $2$ (i.e., from a ${\chi }^{2}$-distribution with $\nu$ degrees of freedom).
One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_students_t (g05snc).

## 4  References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:     dfIntegerInput
On entry: $\nu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df}}\ge 1$.
3:     state[$\mathit{dim}$]IntegerCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization functions nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
4:     x[n]doubleOutput
On exit: the $n$ pseudorandom numbers from the specified Student's $t$-distribution.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
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_INVALID_STATE
On entry, state vector has been corrupted or not initialized.

## 7  Accuracy

Not applicable.

The time taken by nag_rand_students_t (g05snc) increases with $\nu$.

## 9  Example

This example prints five pseudorandom numbers from a Student's $t$-distribution with five degrees of freedom, generated by a single call to nag_rand_students_t (g05snc), after initialization by nag_rand_init_repeatable (g05kfc).

### 9.1  Program Text

Program Text (g05snce.c)

None.

### 9.3  Program Results

Program Results (g05snce.r)