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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 μ$\mu$ and variance σ2${\sigma }^{2}$.

## 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)
 f(x) = 1/(σ×sqrt(2π))exp( − ((x − μ)2)/(2σ2)) . $f(x)=1σ⁢2π exp(- (x-μ) 22σ2 ) .$
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:     n – int64int32nag_int scalar
n$n$, the number of pseudorandom numbers to be generated.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     xmu – double scalar
μ$\mu$, the mean of the distribution.
3:     var – double scalar
σ2${\sigma }^{2}$, the variance of the distribution.
Constraint: var0.0${\mathbf{var}}\ge 0.0$.
4:     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 Normal 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 = 3${\mathbf{ifail}}=3$
On entry, var < 0.0${\mathbf{var}}<0.0$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, state vector was not initialized or has been corrupted.

Not applicable.

None.

## Example

```function nag_rand_dist_normal_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
xmu = 1;
var = 1.5;
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_normal(n, xmu, var, state)
```
```

state =

17
1234
1
0
4110
11820
23399
29340
17917
13895
19930
8
0
1234
1
1
1234

x =

1.4272
-0.5254
1.8109
2.0232
-0.5380

ifail =

0

```
```function g05sk_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
xmu = 1;
var = 1.5;
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sk(n, xmu, var, state)
```
```

state =

17
1234
1
0
4110
11820
23399
29340
17917
13895
19930
8
0
1234
1
1
1234

x =

1.4272
-0.5254
1.8109
2.0232
-0.5380

ifail =

0

```