Burnt Finger Poisson Distribution

Format: BurntFingerPoisson(l1 ,l2)

 


Uses

The Burnt Finger Poisson distribution models the total number of incidents that may occur in a period under the following rules:

    • At the beginning of the period, incidents may occur according to a Poisson process with mean l1;

    • If an incident does occur, then subsequent incidents may occur according to a Poisson process with mean l2;

    •  l2  <  l1.

This type of situation occurs when, for example, an individual has an expected rate of accidents l1, but if an accident occurs the individual will become more careful (his/her ‘fingers got burned’) so that for the rest of the modeled time a new, lower expected accident rate l2  applies.

Looking at the above chart can help better understand the BurntFingerPoisson. The probability of zero events is the same for all three distributions since it is the probability of zero for a Poisson(3). Since Poisson(3) = BurntFingerPoisson(3,3), we can compare the effect of the decrease in  l2 between distributions: the lower the vallue of l2 the more cautious the person has become, so the more likely that just one event will have occurred and the smaller the probability of high numbers of events.

Comments

It can be useful to compare the distribution fits for count data from a variety of Poisson-related distributions like the Poisson, Pólya, Delaporte, ZIPoisson, PoissonUniform and BurntFingerPoisson because they allow you to test different possible mechanisms that may be in place.

It is particularly computationally intensive to simulate from, or calculate probabilities for, the BurntFingerPoisson distribution, especially for large values (e.g. >20) of  l2. A very small amount of approximation has been implemented in ModelRisk for  l2 > 100 to make probability calculations feasible. If possible avoid simulating using the U parameter which can slow down simulating from the BurntFingerPoisson significantly.

 

ModelRisk functions added to Microsoft Excel for the Burnt-Finger Poisson distribution

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

VoseBurntFingerPoissonObject constructs a distribution object for this distribution.

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

VoseBurntFingerPoissonProb10 returns the log10 of the probability mass or cumulative distribution function.

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

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

VoseBurntFingerPoissonFitObject returns the parameters of this distribution fitted to data.

 

Burnt Finger Poisson distribution equations

 

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