Format: VosePoisson(l*t*,
U)

The Poisson distribution is specified by its mean, which is usually denoted l. In this ModelRisk Help File we often use the product l*t as the parameter for the Poisson distribution, because it generally makes understanding and applying Poisson modeling much easier. Two examples of the Poisson distribution are shown below, with lt equal to 5 and 23 respectively:

The Poisson(l*t*) distribution models the number
of occurrences of an event in a time t with an expected rate of
l
events per period t when the time between successive events follows a
Poisson process (we suggest that you read the section on the Poisson
process first, before continuing here).

If *b*
is the mean time between events, as used by the Exponential
distribution, then l = 1/*b*.
For example, imagine that records show that a computer crashes on average
once every 250 hours of operation (*b*=250
hours), then the rate of crashing l is
1/250 crashes per hour. Thus a Poisson *(*1000/250*)* = Poisson(4) distribution models the number of
crashes that could occur in the next 1000 hours of operation.

The Poisson distribution is one of the most important in risk analysis, so you will find a large number of examples. Here are two:

Time series model of events occurring randomly in time;

Fire incidence modeling for integrated risk management

The Poisson distribution is one of several that are use to model claim frequencies in insurance. The distribution mean (l*t) is often referred to as the Poisson intensity.

The Poisson mean intensity may itself be a random variable: for example, insurance claim frequencies will go up and down due to external factors like the occurrence of storms, bad weather, heat waves, etc. The Gamma distribution is most commonly used (because it has flexibility of shape and because the mathematics works out neatly) to model the variability of the Poisson intensity, i.e:

Number of claims = Poisson(Gamma(a,*b*))

which is a Polya distribution. In the special situation where a is an integer it is also a Negative Binomial distribution:

Poisson(Gamma(a,*b*))=
Polya(a,*b*)

Poisson(Gamma(a,*b*))=
NegBin(a,1/(1+*b*)
if a
is an integer

We also have:

Poisson(l+Gamma(a,*b*))=
Delaporte(a,*b*,l)

The Poisson distribution has the useful property: Poisson(*a*)
+ Poisson(*b*) = Poisson(*a*+*b*). This property says in
words that if a accidents are expected to happen in some period and b
in another period, we could estimate the variability of the total number
of accidents in the total period with a Poisson(a + b).

The Poisson distribution is related to the Exponential
and Gamma
distributions, through the Poisson process. The Poisson distribution and
process are named after the French mathematician and physicist *Simeon
Denis Poisson*, though *de Moivre* (1711) derived the distribution
before Poisson.

The Poisson distribution is often thought incorrectly as being applied
only to rare events, perhaps because of the work by *Bortkiewicz*(1898)
who looked at the frequency of infantry deaths in the Prussian Army Corps
from being kicked by a horse, and who described the scenarios in which
the Poisson distribution fits well as the 'Law of Small Numbers'. Bortkiewicz
also fit Poisson distributions to child suicide rates in Prussia. But
a rare event applies some subjective idea of what a 'long time' must be
and the Poisson mathematics works at all scales of time. The difference
between two independent Poisson variables is a Skellam
distribution.

When modeling or analyzing *counting
data*, it is often desirable to modify probability of zero of the
discrete distribution we use, to more accurately model the probability
of "no event occurring". We can make two types of modifications
to our distribution for this:

**Zero-inflated model**- we increase the probability of zero. This model is also known as the ZIP-model. (equations)**Zero-truncated model**- we entirely remove the probability of zero events occurring. (equations)

See also: Zero-modified counting distributions

VosePoisson generates values from this distribution or calculates a percentile.

VosePoissonObject
constructs a distribution object for this distribution. **Professional and Industrial editions only.**

VosePoissonProb
returns the probability mass or cumulative distribution function for this
distribution. **Professional
and Industrial editions only.**

VosePoissonProb10
returns the log10 of the probability mass or cumulative distribution function.
**Professional and
Industrial editions only.**

VosePoissonFit
generates values from this distribution fitted to data, or calculates
a percentile from the fitted distribution. **Professional
and Industrial editions only.**

VosePoissonFitObject
constructs a distribution object of this distribution fitted to data.
**Professional and
Industrial editions only.**

VosePoissonFitP
returns the parameters of this distribution fitted to data. **Professional and Industrial editions only.**