# 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.

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.

GAMMAINC

GNORMAL

GRANDBINOMIAL

GRANDGAMMA

GRANDOM

SEEDRAND

## References:

[1]   Donald E. Knuth

The Art of Computer Programming, Volume 2.