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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_dist_normal (g05sk)

## Purpose

nag_rand_dist_normal (g05sk) generates a vector of pseudorandom numbers taken from a Normal (Gaussian) distribution with mean $\mu$ and variance ${\sigma }^{2}$.

## Syntax

[state, x, ifail] = g05sk(n, xmu, var, state)
[state, x, ifail] = nag_rand_dist_normal(n, xmu, var, state)

## Description

The distribution has PDF (probability distribution function)
 $fx=1σ⁢2π exp- x-μ 22σ2 .$
nag_rand_dist_normal (g05sk) uses the algorithm of Wichura (1988).
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_normal (g05sk).

## References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley
Wichura (1988) Algorithm AS 241: the percentage points of the Normal distribution Appl. Statist. 37 477–484

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{xmu}$ – double scalar
$\mu$, the mean of the distribution.
3:     $\mathrm{var}$ – double scalar
${\sigma }^{2}$, the variance of the distribution.
Constraint: ${\mathbf{var}}\ge 0.0$.
4:     $\mathrm{state}\left(:\right)$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.

### Output Parameters

1:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Contains updated information on the state of the generator.
2:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The $n$ pseudorandom numbers from the specified Normal distribution.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{var}}\ge 0.0$.
${\mathbf{ifail}}=4$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Not applicable.

None.

## Example

This example prints five pseudorandom numbers from a Normal distribution with mean $1.0$ and variance $1.5$, generated by a single call to nag_rand_dist_normal (g05sk), after initialization by nag_rand_init_repeat (g05kf).
```function g05sk_example

fprintf('g05sk example results\n\n');

% Initialize the base generator to a repeatable sequence
seed  = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
genid, subid, seed);

% Number of variates
n = int64(5);

% Parameters
xmu = 1;
var = 1.5;

% Generate variates from a Normal distribution
[state, x, ifail] = g05sk( ...
n, xmu, var, state);

disp('Variates');
disp(x);

```
```g05sk example results

Variates
1.4272
-0.5254
1.8109
2.0232
-0.5380

```