g05 Chapter Contents
g05 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_rand_neg_bin (g05thc)

## 1  Purpose

nag_rand_neg_bin (g05thc) generates a vector of pseudorandom integers from the discrete negative binomial distribution with parameter $m$ and probability $p$ of success at a trial.

## 2  Specification

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

## 3  Description

nag_rand_neg_bin (g05thc) generates $n$ integers ${x}_{i}$ from a discrete negative binomial distribution, where the probability of ${x}_{i}=I$ ($I$ successes before $m$ failures) is
 $Pxi=I= m+I-1! I!m-1! ×pI×1-pm, I=0,1,….$
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_neg_bin (g05thc) with the same parameter value can then use this reference vector to generate further variates.
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_neg_bin (g05thc).

## 4  References

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_neg_bin (g05thc).
${\mathbf{mode}}=\mathrm{Nag_InitializeAndGenerate}$
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: ${\mathbf{n}}\ge 0$.
3:     mIntegerInput
On entry: $m$, the number of failures of the distribution.
Constraint: ${\mathbf{m}}\ge 0$.
4:     pdoubleInput
On entry: $p$, the parameter of the negative binomial distribution representing the probability of success at a single trial.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
5:     r[lr]doubleCommunication Array
On entry: if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, the reference vector from the previous call to nag_rand_neg_bin (g05thc).
If ${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$, r is not referenced and may be NULL.
On exit: if ${\mathbf{mode}}\ne \mathrm{Nag_GenerateWithoutReference}$, the reference vector.
6:     lrIntegerInput
On entry: the dimension of the array r.
Suggested values:
• if ${\mathbf{mode}}\ne \mathrm{Nag_GenerateWithoutReference}$,
${\mathbf{lr}}=28+\left(20×\sqrt{{\mathbf{m}}×{\mathbf{p}}}+30×{\mathbf{p}}\right)/\left(1-{\mathbf{p}}\right)\text{​ approximately}$;
• otherwise ${\mathbf{lr}}=1$.
Constraints:
• if ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$,
$\begin{array}{cc}{\mathbf{lr}}>& \mathrm{int}\left(\begin{array}{c}\frac{{\mathbf{m}}×{\mathbf{p}}+7.15×\sqrt{{\mathbf{m}}×{\mathbf{p}}}+20.15×{\mathbf{p}}}{1-{\mathbf{p}}}+8.5\end{array}\right)\\ \\ & -\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\left(\begin{array}{c}0,\mathrm{int}\left(\begin{array}{c}\frac{{\mathbf{m}}×{\mathbf{p}}-7.15×\sqrt{{\mathbf{m}}×{\mathbf{p}}}}{1-{\mathbf{p}}}\end{array}\right)\end{array}\right)\right)+9\\ \end{array}$;
• if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, lr must remain unchanged from the previous call to nag_rand_neg_bin (g05thc).
7:     state[$\mathit{dim}$]IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to 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.
8:     x[n]IntegerOutput
On exit: the $n$ pseudorandom numbers from the specified negative binomial distribution.
9:     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, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
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
p or m is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Previous value of ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
NE_REAL
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
NE_REF_VEC
On entry, some of the elements of the array r have been corrupted or have not been initialized.

Not applicable.

Not applicable.

None.

## 10  Example

This example prints $20$ pseudorandom integers from a negative binomial distribution with parameters $m=60$ and $p=0.999$, generated by a single call to nag_rand_neg_bin (g05thc), after initialization by nag_rand_init_repeatable (g05kfc).

### 10.1  Program Text

Program Text (g05thce.c)

None.

### 10.3  Program Results

Program Results (g05thce.r)