nag_rand_compd_poisson (g05tkc) generates
$m$ integers
${x}_{j}$, each from a discrete Poisson distribution with mean
${\lambda}_{j}$, where the probability of
${x}_{j}=I$ is
where
The methods used by this function have low set up times and are designed for efficient use when the value of the parameter
$\lambda $ changes during the simulation. For large samples from a distribution with fixed
$\lambda $ using
nag_rand_poisson (g05tjc) to set up and use a reference vector may be more efficient.
When
$\lambda <7.5$ the product of uniforms method is used, see for example
Dagpunar (1988). For larger values of
$\lambda $ an envelope rejection method is used with a target distribution:
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
$\lambda \ge 87$ 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 functions
nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or
nag_rand_init_nonrepeatable (g05kgc) (for a nonrepeatable sequence) must be called prior to the first call to
nag_rand_compd_poisson (g05tkc).
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
 1:
$\mathbf{m}$ – IntegerInput

On entry: $m$, the number of Poisson distributions for which pseudorandom variates are required.
Constraint:
${\mathbf{m}}\ge 1$.
 2:
$\mathbf{vlamda}\left[{\mathbf{m}}\right]$ – const doubleInput

On entry: the means,
${\lambda}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$, of the Poisson distributions.
Constraint:
$0.0\le {\mathbf{vlamda}}\left[\mathit{j}1\right]\le {\mathbf{nag\_max\_integer}}/2.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
 3:
$\mathbf{state}\left[\mathit{dim}\right]$ – 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.
 4:
$\mathbf{x}\left[{\mathbf{m}}\right]$ – IntegerOutput

On exit: the $m$ pseudorandom numbers from the specified Poisson distributions.
 5:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
Not applicable.
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 implementationspecific information.
None.
This example prints ten pseudorandom integers from five Poisson distributions with means
${\lambda}_{1}=0.5$,
${\lambda}_{2}=5$,
${\lambda}_{3}=10$,
${\lambda}_{4}=500$ and
${\lambda}_{5}=1000$. These are generated by ten calls to
nag_rand_compd_poisson (g05tkc), after initialization by
nag_rand_init_repeatable (g05kfc).
None.