g05 Chapter Contents
g05 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_rand_gen_discrete (g05tdc)

## 1  Purpose

nag_rand_gen_discrete (g05tdc) generates a vector of pseudorandom integers from a discrete distribution with a given PDF (probability density function) or CDF (cumulative distribution function) $p$.

## 2  Specification

 #include #include
 void nag_rand_gen_discrete (Nag_ModeRNG mode, Integer n, const double p[], Integer np, Integer ip1, Nag_DiscreteDistrib itype, double r[], Integer lr, Integer state[], Integer x[], NagError *fail)

## 3  Description

nag_rand_gen_discrete (g05tdc) generates a sequence of $n$ integers ${x}_{i}$, from a discrete distribution defined by information supplied in p. This may either be the PDF or CDF of the distribution. A reference vector is first set up to contain the CDF of the distribution in its higher elements, followed by an index.
Setting up the reference vector and subsequent generation of variates can each be performed by separate calls to nag_rand_gen_discrete (g05tdc) or may be combined in a single call.
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_gen_discrete (g05tdc).

## 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_gen_discrete (g05tdc).
${\mathbf{mode}}=\mathrm{Nag_InitializeAndGenerate}$
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:     p[np]const doubleInput
On entry: the PDF or CDF of the distribution.
Constraints:
• $0.0\le {\mathbf{p}}\left[\mathit{i}-1\right]\le 1.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{np}}$;
• if ${\mathbf{itype}}=\mathrm{Nag_PDF}$, $\sum _{\mathit{i}=1}^{{\mathbf{np}}}{\mathbf{p}}\left[\mathit{i}-1\right]=1.0$;
• if ${\mathbf{itype}}=\mathrm{Nag_CDF}$, ${\mathbf{p}}\left[\mathit{i}-1\right]<{\mathbf{p}}\left[j-1\right]\text{, ​}\mathit{i}.
4:     npIntegerInput
On entry: the number of values supplied in p defining the PDF or CDF of the discrete distribution.
Constraint: ${\mathbf{np}}>0$.
5:     ip1IntegerInput
On entry: the value of the variate, a whole number, to which the probability in ${\mathbf{p}}\left[0\right]$ corresponds.
6:     itypeNag_DiscreteDistribInput
On entry: indicates the type of information contained in p.
${\mathbf{itype}}=\mathrm{Nag_PDF}$
p contains a probability distribution function (PDF).
${\mathbf{itype}}=\mathrm{Nag_CDF}$
p contains a cumulative distribution function (CDF).
Constraint: ${\mathbf{itype}}=\mathrm{Nag_PDF}$ or $\mathrm{Nag_CDF}$.
7:     r[lr]doubleCommunication Array
On entry: if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, the reference vector from the previous call to nag_rand_gen_discrete (g05tdc).
On exit: the reference vector.
8:     lrIntegerInput
On entry: the dimension of the array r.
Suggested values:
• if ${\mathbf{mode}}\ne \mathrm{Nag_GenerateWithoutReference}$, ${\mathbf{lr}}=10+1.4×{\mathbf{np}}$ approximately (for optimum efficiency in generating variates);
• otherwise ${\mathbf{lr}}=1$.
Constraints:
• if ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$, ${\mathbf{lr}}\ge {\mathbf{np}}+8$;
• if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, lr should remain unchanged from the previous call to nag_rand_gen_discrete (g05tdc).
9:     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.
10:   x[n]IntegerOutput
On exit: contains $n$ pseudorandom numbers from the specified discrete distribution.
11:   failNagError *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.
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, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{np}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{np}}>0$.
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
The value of np or ip1 is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{np}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{np}}=⟨\mathit{\text{value}}⟩$.
Previous value of ${\mathbf{ip1}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{ip1}}=⟨\mathit{\text{value}}⟩$.
NE_REAL_ARRAY
On entry, at least one element of the vector p is less than $0.0$ or greater than $1.0$.
On entry, ${\mathbf{itype}}=\mathrm{Nag_CDF}$ and the values of p are not all in stricly ascending order.
On entry, ${\mathbf{itype}}=\mathrm{Nag_PDF}$ and the sum of the elements of p do not equal one.
On entry, ${\mathbf{p}}\left[{\mathbf{np}}-1\right]=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{itype}}=\mathrm{Nag_CDF}$, ${\mathbf{p}}\left[{\mathbf{np}}-1\right]=1.0$.
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 variates from a discrete distribution whose PDF, $p$, is defined as follows:
 $n p -5 0.01 -4 0.02 -3 0.04 -2 0.08 -1 0.20 -0 0.30 -1 0.20 -2 0.08 -3 0.04 -4 0.02 -5 0.01$
The reference vector is set up and and the variates are generated by a single call to nag_rand_gen_discrete (g05tdc), after initialization by nag_rand_init_repeatable (g05kfc).

### 10.1  Program Text

Program Text (g05tdce.c)

None.

### 10.3  Program Results

Program Results (g05tdce.r)