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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_dist_lognormal (g05sm)

## Purpose

nag_rand_dist_lognormal (g05sm) generates a vector of pseudorandom numbers from a log-normal distribution with parameters μ$\mu$ and σ2${\sigma }^{2}$.

## Syntax

[state, x, ifail] = g05sm(n, xmu, var, state)
[state, x, ifail] = nag_rand_dist_lognormal(n, xmu, var, state)

## Description

The distribution has PDF (probability density function)
 f(x) = 1/( xσ×sqrt(2π) ) exp( − ((lnx − μ)2)/(2σ2)) if ​ x > 0 , f(x) = 0 otherwise,
$f(x) = 1 xσ⁢2π exp( - (ln⁡x-μ) 2 2σ2 ) if ​ x>0 , f(x)=0 otherwise,$
i.e., lnx$\mathrm{ln}x$ is normally distributed with mean μ$\mu$ and variance σ2${\sigma }^{2}$. nag_rand_dist_lognormal (g05sm) evaluates expyi$\mathrm{exp}{y}_{i}$, where the yi${y}_{i}$ are generated by nag_rand_dist_normal (g05sk) from a Normal distribution with mean μ$\mu$ and variance σ2${\sigma }^{2}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
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_lognormal (g05sm).

## 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

## 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 of lnx$\mathrm{ln}x$.
3:     var – double scalar
σ2${\sigma }^{2}$, the variance of the distribution of lnx$\mathrm{ln}x$.
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 log-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 = 2${\mathbf{ifail}}=2$
On entry, unable to calculate exp(xmu)$\mathrm{exp}\left({\mathbf{xmu}}\right)$ as xmu is too large.
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_lognormal_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 = 2;
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_lognormal(n, xmu, var, state)
```
```

state =

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

x =

4.4515
0.4670
6.9331
8.8597
0.4603

ifail =

0

```
```function g05sm_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 = 2;
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sm(n, xmu, var, state)
```
```

state =

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

x =

4.4515
0.4670
6.9331
8.8597
0.4603

ifail =

0

```