g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rand_binomial (g05tac)

## 1  Purpose

nag_rand_binomial (g05tac) generates a vector of pseudorandom integers from the discrete binomial distribution with parameters $m$ and $p$.

## 2  Specification

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

## 3  Description

nag_rand_binomial (g05tac) generates $n$ integers ${x}_{i}$ from a discrete binomial distribution, where the probability of ${x}_{i}=I$ is
 $Pxi=I= m! I!m-I! ⁢ pI×1-pm-I, I=0,1,…,m,$
where $m\ge 0$ and $0\le p\le 1$. This represents the probability of achieving $I$ successes in $m$ trials when the probability of success at a single trial is $p$.
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_binomial (g05tac) with the same parameter values 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_binomial (g05tac).

## 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_binomial (g05tac).
${\mathbf{mode}}=\mathrm{Nag_InitializeAndGenerate}$
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 trials of the distribution.
Constraint: ${\mathbf{m}}\ge 0$.
4:     pdoubleInput
On entry: $p$, the probability of success of the binomial distribution.
Constraint: $0.0\le {\mathbf{p}}\le 1.0$.
5:     r[lr]doubleCommunication Array
On entry: if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, the reference vector from the previous call to nag_rand_binomial (g05tac).
If ${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$, r is not referenced by nag_rand_binomial (g05tac).
On exit: the reference vector.
6:     lrIntegerInput
On entry: the dimension of the array r.
Suggested values:
• if ${\mathbf{mode}}\ne \mathrm{Nag_GenerateWithoutReference}$, ${\mathbf{lr}}=22+20×\sqrt{{\mathbf{m}}×{\mathbf{p}}×\left(1-{\mathbf{p}}\right)}$;
• otherwise ${\mathbf{lr}}=1$.
Constraints:
• if ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$,
$\begin{array}{lll}{\mathbf{lr}}& >& \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},\mathrm{int}\left[{\mathbf{m}}×{\mathbf{p}}+7.15×\sqrt{{\mathbf{m}}×{\mathbf{p}}×\left(1-{\mathbf{p}}\right)}+1\right]\right)\\ & & -\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(0,\mathrm{int}\left[{\mathbf{m}}×{\mathbf{p}}-7.15×\sqrt{{\mathbf{m}}×{\mathbf{p}}×\left(1-{\mathbf{p}}\right)}-7.15\right]\right)+8\end{array}$;
• if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, lr must remain unchanged from the previous call to nag_rand_binomial (g05tac).
7:     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.
8:     x[n]IntegerOutput
On exit: the $n$ pseudorandom numbers from the specified 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: $0.0\le {\mathbf{p}}\le 1.0$.
NE_REF_VEC
On entry, some of the elements of the array r have been corrupted or have not been initialized.

Not applicable.

None.

## 9  Example

This example prints $20$ pseudorandom integers from a binomial distribution with parameters $m=6000$ and $p=0.8$, generated by a single call to nag_rand_binomial (g05tac), after initialization by nag_rand_init_repeatable (g05kfc).

### 9.1  Program Text

Program Text (g05tace.c)

None.

### 9.3  Program Results

Program Results (g05tace.r)