# GRANDPOISSON

## Purpose:

Generates a Poisson distributed random
series.

## Syntax:

GRANDPOISSON(length,
spacing, lambda)

length |
- |
An integer, the length of the output series. |

spacing |
- |
A real, the spacing (delta x) between points. |

lambda |
- |
Optional. A real, the event
rate where 0 ≤ **lambda**.
Defaults to 1.0. |

## Returns:

A series.

## Example:

grandpoisson(100, 0.01, 2.0)

creates a 100 point, Poisson distributed
random series of values from a population where the event rate is 2.0.

## Example:

x = 0..0.1..30;

lambda = 10;

W1: grandpoisson(50000, 1, lambda);label("Random
Poisson")

W2: hist(W1, 100, "prob");label("Histogram")

W3: exp(x*log(lambda) - lambda - gammaln(x+1));overp(W2,
lred);label("Poisson Distribution")

W1 contains 50000 samples of Poisson distributed
random values with an event rate of 10.

W2 contains a 100 sample normalized histogram
of W1.

W3 compares the distribution of the generated
Poisson random series to the analytic distribution with an event rate
of 10.

## Remarks:

For **lambda** less than 30, GRANDPOISSON
uses a pseudo-random number sampling method due to Knuth [1].
For all other values of **lambda**,
a rejection method due to Atkinson is employed [2].

The probability mass function, *P*(*k*),
for Poisson distributed random values is:

where *k*
is a non-negative integer and *λ*
is the event rate.

For *λ*
the average number of events per interval, *P*(*k*)
is the probability of observing *k*
events in that interval. For example, if one observes on average 3 accidents
per month at a particular traffic intersection, assuming a Poisson model,
the probability of observing exactly 4 accidents in a month is:

The cumulative distribution function *F*(*k*)
is:

where *Q*
is the upper regularized incomplete gamma function implemented by GAMMAINC
and ⌊*k*⌋
is the floor function implemented by FLOOR.

The mean and variance of a Poisson distribution
is *λ.*

## See Also:

GAMMAINC

GNORMAL

GRANDBINOMIAL

GRANDGAMMA

GRANDOM

HISTOGRAM

PDFNORM

PROBN

RAND

RANDN

SEEDRAND

## References:

[1]
Donald E. Knuth

*The Art of Computer Programming, Volume 2.*

Addison
Wesley, 1969

[2]
A.
C. Atkinson

*The Computer Generation of Poisson Random
Variables*

* J*ournal
of the Royal Statistical Society Series C (Applied Statistics)

Vol.
28, No. 1. (1979)