g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rand_poisson (g05tjc)

## 1  Purpose

nag_rand_poisson (g05tjc) generates a vector of pseudorandom integers from the discrete Poisson distribution with mean $\lambda$.

## 2  Specification

 #include #include
 void nag_rand_poisson (Nag_ModeRNG mode, Integer n, double lambda, double r[], Integer lr, Integer state[], Integer x[], NagError *fail)

## 3  Description

nag_rand_poisson (g05tjc) 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_poisson (g05tjc) 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_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_poisson (g05tjc).

## 4  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

## 5  Arguments

1:     modeNag_ModeRNGInput
On entry: a code for selecting the operation to be performed by the function.
${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$
Set up reference vector only.
${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$
Generate variates using reference vector set up in a prior call to nag_rand_poisson (g05tjc).
${\mathbf{mode}}=\mathrm{Nag_InitializeAndGenerate}$
Set up reference vector and generate variates.
${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$
Generate variates without using the reference vector.
Constraint: ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$, $\mathrm{Nag_GenerateFromReference}$, $\mathrm{Nag_InitializeAndGenerate}$ or $\mathrm{Nag_GenerateWithoutReference}$.
2:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
On entry: $\lambda$, the mean of the Poisson distribution.
Constraint: ${\mathbf{lambda}}\ge 0.0$.
4:     r[lr]doubleCommunication Array
On entry: if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, the reference vector from the previous call to nag_rand_poisson (g05tjc).
If ${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$, r is not referenced by nag_rand_poisson (g05tjc).
On exit: the reference vector.
5:     lrIntegerInput
On entry: the dimension of the array r.
Suggested values:
• if ${\mathbf{mode}}\ne \mathrm{Nag_GenerateWithoutReference}$, ${\mathbf{lr}}=30+20×\sqrt{{\mathbf{lambda}}}+{\mathbf{lambda}}$;
• otherwise ${\mathbf{lr}}=1$.
Constraints:
• if ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$,
• if $\sqrt{{\mathbf{lambda}}}>7.15$, ${\mathbf{lr}}>9+\mathrm{int}\left(8.5+14.3×\sqrt{{\mathbf{lambda}}}\right)$;
• otherwise ${\mathbf{lr}}>9+\mathrm{int}\left({\mathbf{lambda}}+7.15×\sqrt{{\mathbf{lambda}}}+8.5\right)$;
• if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, lr must remain unchanged from the previous call to nag_rand_poisson (g05tjc).
6:     state[$\mathit{dim}$]IntegerCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization functions nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
7:     x[n]IntegerOutput
On exit: the $n$ pseudorandom numbers from the specified Poisson distribution.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, lr is too small when ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$: ${\mathbf{lr}}=〈\mathit{\text{value}}〉$, minimum length required $\text{}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_PREV_CALL
lambda is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{lambda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{lambda}}=〈\mathit{\text{value}}〉$.
NE_REAL
lambda is such that lr would have to be larger than the largest representable integer. Use ${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$ instead. ${\mathbf{lambda}}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{lambda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lambda}}\ge 0.0$.
NE_REF_VEC
On entry, some of the elements of the array r have been corrupted or have not been initialized.

Not applicable.

None.

## 9  Example

This example prints $10$ pseudorandom integers from a Poisson distribution with mean $\lambda =20$, generated by a single call to nag_rand_poisson (g05tjc), after initialization by nag_rand_init_repeatable (g05kfc).

### 9.1  Program Text

Program Text (g05tjce.c)

None.

### 9.3  Program Results

Program Results (g05tjce.r)