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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_dist_expmix (g05sg)

## Purpose

nag_rand_dist_expmix (g05sg) generates a vector of pseudorandom numbers from an exponential mix distribution composed of $m$ exponential distributions each having a mean ${a}_{i}$ and weight ${w}_{i}$.

## Syntax

[state, x, ifail] = g05sg(n, a, wgt, state, 'nmix', nmix)
[state, x, ifail] = nag_rand_dist_expmix(n, a, wgt, state, 'nmix', nmix)

## Description

The distribution has PDF (probability density function)
 $fx = ∑i=1m 1ai wi e-x/ai if ​x≥0, fx = 0 otherwise,$
where $\sum _{i=1}^{m}{w}_{i}=1$ and ${a}_{i}>0$, ${w}_{i}\ge 0$.
nag_rand_dist_expmix (g05sg) returns the values ${x}_{i}$ by selecting, with probability ${w}_{j}$, random variates from an exponential distribution with argument ${a}_{j}$.
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_expmix (g05sg).

## 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:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{a}\left({\mathbf{nmix}}\right)$ – double array
The $m$ parameters ${a}_{i}$ for the $m$ exponential distributions in the mix.
Constraint: ${\mathbf{a}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{nmix}}$.
3:     $\mathrm{wgt}\left({\mathbf{nmix}}\right)$ – double array
The $m$ weights ${w}_{i}$ for the $m$ exponential distributions in the mix.
Constraints:
• $\sum _{\mathit{i}=1}^{m}{\mathbf{wgt}}\left(\mathit{i}\right)=1.0$;
• ${\mathbf{wgt}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,m$.
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.

### Optional Input Parameters

1:     $\mathrm{nmix}$int64int32nag_int scalar
Default: the dimension of the arrays a, wgt. (An error is raised if these dimensions are not equal.)
$m$, the number of exponential distributions in the mix.
Constraint: ${\mathbf{nmix}}\ge 1$.

### 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 exponential mix 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: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{nmix}}\ge 1$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{a}}\left(i\right)>0.0$.
${\mathbf{ifail}}=4$
Constraint: sum of ${\mathbf{wgt}}=1.0$.
Constraint: ${\mathbf{wgt}}\left(i\right)\ge 0.0$.
${\mathbf{ifail}}=5$
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 the first five pseudorandom numbers from an exponential mix distribution comprising three exponential distributions with parameters ${a}_{1}=1.0$, ${a}_{2}=5.0$ and ${a}_{3}=2.0$, and with respective weights $0.5$, $0.3$ and $0.2$. The numbers are generated by a single call to nag_rand_dist_expmix (g05sg), after initialization by nag_rand_init_repeat (g05kf).
```function g05sg_example

fprintf('g05sg 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
a   = [1;   5;   2];
wgt = [0.5; 0.3; 0.2];

% Generate variates from an exponential mix distribution
[state, x, ifail] = g05sg( ...
n, a, wgt, state);

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

```
```g05sg example results

Variates
0.4520
2.2398
1.4649
0.2253
11.2884

```