g05 Chapter Contents
g05 Chapter Introduction
NAG 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:    $\mathbf{mode}$Nag_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:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of trials of the distribution.
Constraint: ${\mathbf{m}}\ge 0$.
4:    $\mathbf{p}$doubleInput
On entry: $p$, the probability of success of the binomial distribution.
Constraint: $0.0\le {\mathbf{p}}\le 1.0$.
5:    $\mathbf{r}\left[{\mathbf{lr}}\right]$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 and may be NULL.
On exit: if ${\mathbf{mode}}\ne \mathrm{Nag_GenerateWithoutReference}$, the reference vector.
6:    $\mathbf{lr}$IntegerInput
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:    $\mathbf{state}\left[\mathit{dim}\right]$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:    $\mathbf{x}\left[{\mathbf{n}}\right]$IntegerOutput
On exit: the $n$ pseudorandom numbers from the specified binomial distribution.
9:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
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.

## 8  Parallelism and Performance

nag_rand_binomial (g05tac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10  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).

### 10.1  Program Text

Program Text (g05tace.c)

None.

### 10.3  Program Results

Program Results (g05tace.r)