g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rngs_neg_bin (g05mcc)

## 1  Purpose

nag_rngs_neg_bin (g05mcc) 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_rngs_neg_bin (Integer mode, Integer m, double p, Integer n, Integer x[], Integer igen, Integer iseed[], double r[], NagError *fail)

## 3  Description

nag_rngs_neg_bin (g05mcc) 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_rngs_neg_bin (g05mcc) with the same parameter value can then use this reference vector to generate further variates.
One of the initialization functions nag_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_neg_bin (g05mcc).

## 4  References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 5  Arguments

1:     modeIntegerInput
On entry: 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_rngs_neg_bin (g05mcc).
${\mathbf{mode}}=2$
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:     mIntegerInput
On entry: $m$, the number of failures of the distribution.
Constraint: ${\mathbf{m}}\ge 0$.
3:     pdoubleInput
On entry: $p$, the parameter of the negative binomial distribution representing the probability of success at a single trial.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
4:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 1$.
5:     x[n]IntegerOutput
On exit: the $n$ pseudorandom numbers from the specified negative binomial distribution.
6:     igenIntegerInput
On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).
7:     iseed[$4$]IntegerCommunication Array
On entry: contains values which define the current state of the selected generator.
On exit: contains updated values defining the new state of the selected generator.
8:     r[$\mathit{dim}$]doubleCommunication Array
Note: the dimension, dim, of the array r must be at least
• $20+\left(20\sqrt{{\mathbf{m}}×{\mathbf{p}}}+30×{\mathbf{p}}\right)/\left(1-{\mathbf{p}}\right)$ when ${\mathbf{mode}}\ne 3$;
• $1$ otherwise.
On entry: if ${\mathbf{mode}}=1$, the reference vector from the previous call to nag_rngs_neg_bin (g05mcc).
On exit: the reference vector.
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_DIM_INFEASIBLE
p and m are such that the reference vector length would exceed integer range. We recommend setting ${\mathbf{mode}}=3$.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{mode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
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_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, ${\mathbf{p}}<0.0$ or ${\mathbf{p}}\ge 1.0$: ${\mathbf{p}}=〈\mathit{\text{value}}〉$.

Not applicable.

None.

## 9  Example

This example prints five pseudorandom integers from a negative binomial distribution with parameters $m=60$ and $p=0.999$, generated by a single call to nag_rngs_neg_bin (g05mcc), after initialization by nag_rngs_init_repeatable (g05kbc).

### 9.1  Program Text

Program Text (g05mcce.c)

None.

### 9.3  Program Results

Program Results (g05mcce.r)