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# NAG Toolbox: nag_rand_int_poisson (g05tj)

## Purpose

nag_rand_int_poisson (g05tj) generates a vector of pseudorandom integers from the discrete Poisson distribution with mean $\lambda$.

## Syntax

[r, state, x, ifail] = g05tj(mode, n, lambda, r, state)
[r, state, x, ifail] = nag_rand_int_poisson(mode, n, lambda, r, state)

## Description

nag_rand_int_poisson (g05tj) generates $n$ integers ${x}_{i}$ from a discrete Poisson distribution with mean $\lambda$, where the probability of ${x}_{i}=I$ is
 $Pxi=I= λI×e-λ I! , I=0,1,…,$
where $\lambda \ge 0$.
The variates can be generated with or without using a search table and index. If a search table is used then it is stored with the index in a reference vector and subsequent calls to nag_rand_int_poisson (g05tj) with the same parameter values can then use this reference vector to generate further variates. The reference array is found using a recurrence relation if $\lambda$ is less than $50$ and by Stirling's formula otherwise.
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_int_poisson (g05tj).

## 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:     $\mathrm{mode}$int64int32nag_int scalar
A code for selecting the operation to be performed by the function.
${\mathbf{mode}}=0$
Set up reference vector only.
${\mathbf{mode}}=1$
Generate variates using reference vector set up in a prior call to nag_rand_int_poisson (g05tj).
${\mathbf{mode}}=2$
Set up reference vector and generate variates.
${\mathbf{mode}}=3$
Generate variates without using the reference vector.
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
2:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathrm{lambda}$ – double scalar
$\lambda$, the mean of the Poisson distribution.
Constraint: ${\mathbf{lambda}}\ge 0.0$.
4:     $\mathrm{r}\left(\mathit{lr}\right)$ – double array
lr, the dimension of the array, must satisfy the constraint
• if ${\mathbf{mode}}=0$ or $2$,
• if $\sqrt{{\mathbf{lambda}}}>7.15$, $\mathit{lr}>9+\mathrm{int}\left(8.5+14.3×\sqrt{{\mathbf{lambda}}}\right)$;
• otherwise $\mathit{lr}>9+\mathrm{int}\left({\mathbf{lambda}}+7.15×\sqrt{{\mathbf{lambda}}}+8.5\right)$;
• if ${\mathbf{mode}}=1$, lr must remain unchanged from the previous call to nag_rand_int_poisson (g05tj).
If ${\mathbf{mode}}=1$, the reference vector from the previous call to nag_rand_int_poisson (g05tj).
If ${\mathbf{mode}}=3$, r is not referenced.
5:     $\mathrm{state}\left(:\right)$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:     $\mathrm{r}\left(\mathit{lr}\right)$ – double array
If ${\mathbf{mode}}\ne 3$, the reference vector.
2:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Contains updated information on the state of the generator.
3:     $\mathrm{x}\left({\mathbf{n}}\right)$int64int32nag_int array
The $n$ pseudorandom numbers from the specified Poisson distribution.
4:     $\mathrm{ifail}$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: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{lambda}}\ge 0.0$.
lambda is such that lr would have to be larger than the largest representable integer.
${\mathbf{ifail}}=4$
lambda is not the same as when r was set up in a previous call.
On entry, some of the elements of the array r have been corrupted or have not been initialized.
${\mathbf{ifail}}=5$
On entry, lr is too small when ${\mathbf{mode}}=0$ or $2$.
${\mathbf{ifail}}=6$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
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 $10$ pseudorandom integers from a Poisson distribution with mean $\lambda =20$, generated by a single call to nag_rand_int_poisson (g05tj), after initialization by nag_rand_init_repeat (g05kf).
```function g05tj_example

fprintf('g05tj 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(10);

% Parameters
lambda = 20;

% Generate variates from a Poisson distribution
mode = int64(2);
r = zeros(120, 1);
[r, state, x, ifail] = g05tj( ...
mode, n, lambda, r, state);

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

```
```g05tj example results

Variates
21
15
23
24
14
20
19
23
20
22

```

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