g05tk generates a vector of pseudorandom integers, each from a discrete Poisson distribution with differing parameter.

Syntax

C#
public static void g05tk( int m, double[] vlamda, G05..::..G05State g05state, int[] x, out int ifail )

Visual Basic
Public Shared Sub g05tk ( _ m As Integer, _ vlamda As Double(), _ g05state As G05..::..G05State, _ x As Integer(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void g05tk( int m, array<double>^ vlamda, G05..::..G05State^ g05state, array<int>^ x, [OutAttribute] int% ifail )

F#
static member g05tk : m : int * vlamda : float[] * g05state : G05..::..G05State * x : int[] * ifail : int byref -> unit

Parameters

m: Type: System..::..Int32
On entry: $m$ , the number of Poisson distributions for which pseudorandom variates are required.

Constraint: $m \geq 1$ .

vlamda: Type: array<System..::..Double>[]()[][]
An array of size [m]
On entry: the means, $λ_{j}$ , for $j = 1, 2, \dots, m$ , of the Poisson distributions.

Constraint: $0.0 \leq vlamda [j - 1] \leq x02bb / 2.0$ , for $j = 1, 2, \dots, m$ .

g05state: Type: NagLibrary..::..G05..::..G05State
An Object of type G05.G05State.

x: Type: array<System..::..Int32>[]()[][]
An array of size [m]
On exit: the $m$ pseudorandom numbers from the specified Poisson distributions.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

g05tk generates

m

integers

x_{j}

, each from a discrete Poisson distribution with mean

λ_{j}

, where the probability of

x_{j} = I

P (x_{j} = I) = \frac{λ_{j}^{I} \times e^{- λ_{j}}}{I!}, I = 0, 1, \dots,

where

λ_{j} \geq 0, j = 1, 2, \dots, m .

The methods used by this method have low set up times and are designed for efficient use when the value of the parameter

λ

changes during the simulation. For large samples from a distribution with fixed

λ

using g05tj to set up and use a reference vector may be more efficient.

When

λ < 7.5

the product of uniforms method is used, see for example Dagpunar (1988). For larger values of

λ

an envelope rejection method is used with a target distribution:

\begin{matrix} f (x) = \frac{1}{3} & if ​ |x| \leq 1, \\ f (x) = \frac{1}{3} {|x|}^{- 3} & otherwise. \end{matrix}

This distribution is generated using a ratio of uniforms method. A similar approach has also been suggested by Ahrens and Dieter (1989). The basic method is combined with quick acceptance and rejection tests given by Maclaren (1990). For values of

λ \geq 87

Stirling's approximation is used in the computation of the Poisson distribution function, otherwise tables of factorials are used as suggested by Maclaren (1990).

One of the initialization methods (G05KFF not in this release) (for a repeatable sequence if computed sequentially) or (G05KGF not in this release) (for a non-repeatable sequence) must be called prior to the first call to g05tk.

References

Ahrens J H and Dieter U (1989) A convenient sampling method with bounded computation times for Poisson distributions Amer. J. Math. Management Sci. 1–13

Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press

Maclaren N M (1990) A Poisson random number generator Personal Communication

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$: On entry, $m < 1$ .

$ifail = 2$: On entry, $vlamda [j - 1] < 0.0$ for at least one value of $j$ .
On entry, $2 \times vlamda [j - 1] > x02bb ()$ for at least one value of $j$ .