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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_int_binomial (g05ta)

## Purpose

nag_rand_int_binomial (g05ta) generates a vector of pseudorandom integers from the discrete binomial distribution with parameters m$m$ and p$p$.

## Syntax

[r, state, x, ifail] = g05ta(mode, n, m, p, r, state)
[r, state, x, ifail] = nag_rand_int_binomial(mode, n, m, p, r, state)

## Description

nag_rand_int_binomial (g05ta) generates n$n$ integers xi${x}_{i}$ from a discrete binomial distribution, where the probability of xi = I${x}_{i}=I$ is
 P(xi = I) = (m ! )/(I ! (m − I) ! ) pI × (1 − p)m − I,  I = 0,1, … ,m, $P(xi=I)= m! I!(m-I)! ⁢ pI×(1-p)m-I, I=0,1,…,m,$
where m0$m\ge 0$ and 0p1$0\le p\le 1$. This represents the probability of achieving I$I$ successes in m$m$ trials when the probability of success at a single trial is p$p$.
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_int_binomial (g05ta) with the same parameter values can then use this reference vector to generate further variates.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_int_binomial (g05ta).

## 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

## Parameters

### Compulsory Input Parameters

1:     mode – int64int32nag_int scalar
A code for selecting the operation to be performed by the function.
mode = 0${\mathbf{mode}}=0$
Set up reference vector only.
mode = 1${\mathbf{mode}}=1$
Generate variates using reference vector set up in a prior call to nag_rand_int_binomial (g05ta).
mode = 2${\mathbf{mode}}=2$
Set up reference vector and generate variates.
mode = 3${\mathbf{mode}}=3$
Generate variates without using the reference vector.
Constraint: mode = 0${\mathbf{mode}}=0$, 1$1$, 2$2$ or 3$3$.
2:     n – int64int32nag_int scalar
n$n$, the number of pseudorandom numbers to be generated.
Constraint: n0${\mathbf{n}}\ge 0$.
3:     m – int64int32nag_int scalar
m$m$, the number of trials of the distribution.
Constraint: m0${\mathbf{m}}\ge 0$.
4:     p – double scalar
p$p$, the probability of success of the binomial distribution.
Constraint: 0.0p1.0$0.0\le {\mathbf{p}}\le 1.0$.
5:     r(lr) – double array
lr, the dimension of the array, must satisfy the constraint
• if mode = 0${\mathbf{mode}}=0$ or 2$2$,
 lr > min (m,int[m × p + 7.15 × sqrt( m × p × (1 − p) ) + 1]) − max (0,int[m × p − 7.15 × sqrt( m × p × (1 − p) ) − 7.15]) + 8
$\begin{array}{lll}\mathit{lr}& >& \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},\mathrm{int}\left[{\mathbf{m}}×{\mathbf{p}}+7.15×\sqrt{{\mathbf{m}}×{\mathbf{p}}×\left(1-{\mathbf{p}}\right)}+1\right]\right)\\ & & -\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(0,\mathrm{int}\left[{\mathbf{m}}×{\mathbf{p}}-7.15×\sqrt{{\mathbf{m}}×{\mathbf{p}}×\left(1-{\mathbf{p}}\right)}-7.15\right]\right)+8\end{array}$;
• if mode = 1${\mathbf{mode}}=1$, lr must remain unchanged from the previous call to nag_rand_int_binomial (g05ta).
If mode = 1${\mathbf{mode}}=1$, the reference vector from the previous call to nag_rand_int_binomial (g05ta).
If mode = 3${\mathbf{mode}}=3$, r is not referenced by nag_rand_int_binomial (g05ta).
6:     state( : $:$) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

None.

lr

### Output Parameters

1:     r(lr) – double array
The reference vector.
2:     state( : $:$) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains updated information on the state of the generator.
3:     x(n) – int64int32nag_int array
The n$n$ pseudorandom numbers from the specified binomial distribution.
4:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, mode0${\mathbf{mode}}\ne 0$, 1$1$, 2$2$ or 3$3$.
ifail = 2${\mathbf{ifail}}=2$
On entry, n < 0${\mathbf{n}}<0$.
ifail = 3${\mathbf{ifail}}=3$
On entry, m < 0${\mathbf{m}}<0$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, p < 0.0${\mathbf{p}}<0.0$, or p > 1.0${\mathbf{p}}>1.0$.
ifail = 5${\mathbf{ifail}}=5$
On entry, m or p is not the same as when r was set up in a previous call to nag_rand_int_binomial (g05ta) with mode = 0${\mathbf{mode}}=0$ or 2$2$.
On entry, the r vector was not initialized correctly or has been corrupted.
ifail = 6${\mathbf{ifail}}=6$
On entry, lr is too small when mode = 0${\mathbf{mode}}=0$ or 2$2$.
ifail = 7${\mathbf{ifail}}=7$
 On entry, state vector was not initialized or has been corrupted.

Not applicable.

None.

## Example

function nag_rand_int_binomial_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
mode = int64(2);
n = int64(20);
m = int64(6000);
p = 0.8;
r = zeros(6007, 1);
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[r, state, x, ifail] = nag_rand_int_binomial(mode, n, m, p, r, state);
x, ifail

x =

4811
4761
4821
4826
4761
4800
4791
4825
4800
4814
4749
4780
4810
4750
4807
4782
4778
4877
4840
4802

ifail =

0

function g05ta_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
mode = int64(2);
n = int64(20);
m = int64(6000);
p = 0.8;
r = zeros(6007, 1);
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[r, state, x, ifail] = g05ta(mode, n, m, p, r, state);
x, ifail

x =

4811
4761
4821
4826
4761
4800
4791
4825
4800
4814
4749
4780
4810
4750
4807
4782
4778
4877
4840
4802

ifail =

0