Poisson distribution



Format: Poisson(lt)

Poisson equations

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:

Uses

The Poisson(lt) 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).

Examples

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.

Comments

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.

Zero-modified version

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.

  • Zero-truncated model - we entirely remove the probability of zero events occurring.

ModelRisk functions added to Microsoft Excel for the Poisson distribution

VosePoisson generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VosePoissonObject constructs a distribution object for this distribution.

VosePoissonProb returns the probability mass or cumulative distribution function for this distribution.

VosePoissonProb10 returns the log10 of the probability mass or cumulative distribution function.  

VosePoissonFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VosePoissonFitObject constructs a distribution object of this distribution fitted to data.

VosePoissonFitP returns the parameters of this distribution fitted to data.

 

ModelRisk functions added to Microsoft Excel for the Zero-Inflated Poisson distribution

VoseZIPoisson generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZIPoissonObject constructs a distribution object for this distribution.

VoseZIPoissonProb returns the probability mass or cumulative distribution function for this distribution.

VoseZIPoissonProb10 returns the log10 of the probability mass or cumulative distribution function.  

VoseZIPoissonFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseZIPoissonFitObject constructs a distribution object of this distribution fitted to data.

VoseZIPoissonFitP returns the parameters of this distribution fitted to data.

 

ModelRisk functions added to Microsoft Excel for the Zero-Truncated Poisson distribution

VoseZTPoisson generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZTPoissonObject constructs a distribution object for this distribution.

VoseZTPoissonProb returns the probability mass or cumulative distribution function for this distribution.

VoseZTPoissonProb10 returns the log10 of the probability mass or cumulative distribution function.  

VoseZTPoissonFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseZTPoissonFitObject constructs a distribution object of this distribution fitted to data.

VoseZTPoissonFitP returns the parameters of this distribution fitted to data.

 

Poisson distribution equations

Zero-Inflated Poisson distribution equations

Zero-Truncated Poisson distribution equations

 

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