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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_dist_students_t (g05sn)

## Purpose

nag_rand_dist_students_t (g05sn) generates a vector of pseudorandom numbers taken from a Student's t$t$-distribution with ν$\nu$ degrees of freedom.

## Syntax

[state, x, ifail] = g05sn(n, df, state)
[state, x, ifail] = nag_rand_dist_students_t(n, df, state)

## Description

The distribution has PDF (probability density function)
 f(x) = ( ((ν − 1)/2) ! )/(((1/2)ν − 1) ! sqrt(πν) (1 + (x2)/ν)(1/2)(ν + 1)). $f(x)= (ν-12) ! (12ν-1)!πν (1+x2ν) 12(ν+1) .$
nag_rand_dist_students_t (g05sn) calculates the values
 yisqrt(ν/(zi)),   i = 1, … ,n $yiνzi, i= 1,…,n$
where the yi${y}_{i}$ are generated by nag_rand_dist_normal (g05sk) from a Normal distribution with mean 0$0$ and variance 1.0$1.0$, and the zi${z}_{i}$ are generated by nag_rand_dist_gamma (g05sj) from a gamma distribution with parameters (1/2)ν$\frac{1}{2}\nu$ and 2$2$ (i.e., from a χ2${\chi }^{2}$-distribution with ν$\nu$ degrees of freedom).
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_dist_students_t (g05sn).

## References

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

## Parameters

### Compulsory Input Parameters

1:     n – int64int32nag_int scalar
n$n$, the number of pseudorandom numbers to be generated.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     df – int64int32nag_int scalar
ν$\nu$, the number of degrees of freedom of the distribution.
Constraint: df1 ${\mathbf{df}}\ge 1$.
3:     state( : $:$) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

None.

None.

### Output Parameters

1:     state( : $:$) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains updated information on the state of the generator.
2:     x(n) – double array
The n$n$ pseudorandom numbers from the specified Student's t$t$-distribution.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, n < 0${\mathbf{n}}<0$.
ifail = 2${\mathbf{ifail}}=2$
On entry, df < 1${\mathbf{df}}<1$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, state vector was not initialized or has been corrupted.

## Accuracy

Not applicable.

The time taken by nag_rand_dist_students_t (g05sn) increases with ν$\nu$.

## Example

```function nag_rand_dist_students_t_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
df = int64(5);
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_students_t(n, df, state)
```
```

state =

17
1234
1
0
6694
27818
10435
15383
17917
13895
19930
8
0
1234
1
1
1234

x =

0.3849
-0.9461
-2.2814
0.1127
0.5272

ifail =

0

```
```function g05sn_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
df = int64(5);
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sn(n, df, state)
```
```

state =

17
1234
1
0
6694
27818
10435
15383
17917
13895
19930
8
0
1234
1
1
1234

x =

0.3849
-0.9461
-2.2814
0.1127
0.5272

ifail =

0

```