This distribution is due to Skellam (1946 [SKE1]) and is a generalization of a problem considered by Irwin (1937 [IRW1]). It arises as the distribution of the difference between two random variates that follow a Poisson distribution with mean values m1 and m2 respectively (Irwin considered the case where the two means were equal). Skellam shows that the distribution is of the form:

which can also be written in terms of Bessel functions. Clearly if m1=m2 the mean of the distribution will be zero and the distribution will be symmetric about zero.

Although not widely applied, the Skellam distribution has been used in fields as varied as physics, medicine and sports data. For the case where the data is drawn from Poisson distributions with the same mean values, which are not small, the distribution is symmetric and can be reasonably well approximated by the unit Normal distribution using the adjusted variable:

The distribution is very simple to simulate using the widely available facilities in many software packages for the generation of random Poisson variates. However the distribution itself is not generally available, so must be created programmatically in most instances.

Key measures for the Skellam distribution are provided in the table below:

Mean |
m1-m2 |
---|---|

Variance |
m1+m2 |

Skewness |
(m1-m2)/(m1+m2)3/2 |

Kurtosis |
1/(m1+m2) |

MGF |

References

[JOH1] Johnson N L, Kotz S (1969) Discrete distributions. Houghton Mifflin/J Wiley & Sons, New York

[IRW1] Irwin J O (1937) The frequency distribution of the difference between two independent variates following the same Poisson distribution. J. of the Royal Statistical Society, Series A, 100, 415-416

[SKE1] Skellam J G (1946) The frequency distribution of the difference between two Poisson variates belonging to different populations. J. of the Royal Statistical Society, Series A, 109, 296

Wikipedia: Skellam distribution: http://en.wikipedia.org/wiki/Skellam_distribution