NAG C Library Function Document

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 <nag.h>

#include <nagg05.h>

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: $mode$ – Nag_ModeRNGInput

On entry: a code for selecting the operation to be performed by the function.

$mode = Nag_InitializeReference$: Set up reference vector only.
$mode = Nag_GenerateFromReference$: Generate variates using reference vector set up in a prior call to nag_rand_gen_discrete (g05tdc).
$mode = Nag_InitializeAndGenerate$: Set up reference vector and generate variates.
$mode = Nag_GenerateWithoutReference$: Generate variates without using the reference vector.

Constraint:

mode = Nag_InitializeReference

Nag_GenerateFromReference

Nag_InitializeAndGenerate

Nag_GenerateWithoutReference

2: $n$ – IntegerInput

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 0

3: $p [np]$ – const doubleInput

On entry: the PDF or CDF of the distribution.

Constraints:

$0.0 \leq p [i - 1] \leq 1.0$ , for $i = 1, 2, \dots, np$ ;
if $itype = Nag_PDF$ , $\sum_{i = 1}^{np} p [i - 1] = 1.0$ ;
if $itype = Nag_CDF$ , $p [i - 1] < p [j - 1], i < j and p [np - 1] = 1.0$ .

4: $np$ – IntegerInput

On entry: the number of values supplied in p defining the PDF or CDF of the discrete distribution.

Constraint:

np > 0

5: $ip1$ – IntegerInput

On entry: the value of the variate, a whole number, to which the probability in

p [0]

corresponds.

6: $itype$ – Nag_DiscreteDistribInput

On entry: indicates the type of information contained in p.

$itype = Nag_PDF$: p contains a probability distribution function (PDF).
$itype = Nag_CDF$: p contains a cumulative distribution function (CDF).

Constraint:

itype = Nag_PDF

Nag_CDF

7: $r [lr]$ – doubleCommunication Array

On entry: if

mode = Nag_GenerateFromReference

, the reference vector from the previous call to nag_rand_gen_discrete (g05tdc).

On exit: the reference vector.

8: $lr$ – IntegerInput

On entry: the dimension of the array r.

Suggested values:

if $mode \neq Nag_GenerateWithoutReference$ , $lr = 10 + 1.4 \times np$ approximately (for optimum efficiency in generating variates);
otherwise $lr = 1$ .

Constraints:

if $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ , $lr \geq np + 8$ ;
if $mode = Nag_GenerateFromReference$ , lr should remain unchanged from the previous call to nag_rand_gen_discrete (g05tdc).

9: $state [\dim]$ – IntegerCommunication Array

Note: the dimension,

\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: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, lr is too small when $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ : $lr = 〈value〉$ , minimum length required $= 〈value〉$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $np = 〈value〉$ .
Constraint: $np > 0$ .
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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 $np = 〈value〉$ and $np = 〈value〉$ .
Previous value of $ip1 = 〈value〉$ and $ip1 = 〈value〉$ .
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, $itype = Nag_CDF$ and the values of p are not all in stricly ascending order.

On entry, $itype = Nag_PDF$ and the sum of the elements of p do not equal one.

On entry, $p [np - 1] = 〈value〉$ .
Constraint: if $itype = Nag_CDF$ , $p [np - 1] = 1.0$ .
NE_REF_VEC: On entry, some of the elements of the array r have been corrupted or have not been initialized.

7

Accuracy

Not applicable.

8

Parallelism and Performance

nag_rand_gen_discrete (g05tdc) 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.

9

Further Comments

None.

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Example

This example prints

20

pseudorandom variates from a discrete distribution whose PDF,

p

, is defined as follows:

\begin{matrix} 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 \end{matrix}

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).