Skellam distribution | Vose Software

# Skellam distribution

Format: Skellam(
λ1, λ2)

A Skellam(
λ1, λ2) distribution returns discrete values between -infinity and +infinity. This is a unique property among discrete distributions. Examples of the Skellam distribution are shown below:

## Uses

The Skellam distribution models the difference between two independent Poisson distributed variables as follows:

Skellam(λ1, λ2) = Poisson(λ1) - Poisson(λ2)

The Skellam distribution has a number of uses, essentially relating to the above formula. For example the difference in number of accidents (or murders, strikes, catastrophes, etc. – anything that occurs randomly in time) between, for example, two cities, two populations, two years, etc – where it is assumed that the expected rate of occurrence for the two variables are λ1 and λ2.

For the sports and games fans among you, it can be used to model the point spread between two teams where points are scored independently and in single units (like field hockey, ice hockey, soccer, chess, backgammon (if you ignore doubling) and baseball), but not where points are scored in ‘clusters’ (like rugby where a trial (5 points) may be followed by a conversion (2 points), cricket (1-6 runs from a hit), basketball where a basket is worth 1, 2 or 3 points depending on where the shot was taken from, or American football where a touchdown is worth 5 points, a try is worth 1 or 2 points, a field goal is worth 3 points, and a safety is worth 2 points).

The Skellam distribution first appeared in Skellam (1946).

## Zero-inflated version

When modeling or analyzing counting data, it is often desirable to modify the probability of zero of the discrete distribution we use, to more accurately model the probability of "no event occurring".  A zero-inflated Skellam distribution (increase the probability of zero) is also available in ModelRisk.

## Reference

Skellam, J. G. (1946) "The frequency distribution of the difference between two Poisson variates belonging to different populations". Journal of the Royal Statistical Society, Series A, 109 (3), 296.

## ModelRisk functions added to Microsoft Excel for the Skellam  distribution

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

VoseSkellamObject constructs a distribution object for this distribution.

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

VoseSkellamProb10 returns the log10 of the probability mass or cumulative distribution function.

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

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

VoseSkellamFitP returns the parameters of this distribution fitted to data.

## ModelRisk functions added to Microsoft Excel for the Zero-inflated Skellam distribution

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

VoseZISkellamObject constructs a distribution object for this distribution.

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

VoseZISkellamProb10 returns the log10 of the probability mass or cumulative distribution function.

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

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

VoseZISkellamFitP returns the parameters of this distribution fitted to data.