# Skellam distribution

The Skellam distribution, also known as the Poisson difference distribution, compares counts from two Poisson random variables and represents the difference between them. This two-parameter discrete probability distribution can be applied to independent variables or dependent variables when there is a common additive random contribution that is cancelled by differencing.

J.G. Skellam introduced the Skellam distribution in 1946  as a means of modeling the difference between two Poisson variates that belong to different populations. He discovered that this difference follows a specific distribution, now known as the Skellam distribution, while observing the variance between the number of births and deaths in a population.

## Skellam distribution properties

The Skellam distribution is closed under summation, is relatively easy to sample from, and approaches a normal distribution with larger variance. Examples of the Skellam distribution PMF. Lines connect the discrete data points and do not imply continuity.

The probability mass function (PMF) of the Skellam distribution is :

Where:

• k is the difference between two Poisson random variables; k ∈ {…, -2, -1, 0, 1, 2, …}.
• μ1 and μ2 are expected values or means of two Poisson distributions.
• Ik (z) is the modified Bessel function of the first kind. Since k is an integer, Ik(z)=I|k|(z).

Mean: μ1 – μ2

Variance: μ1 + μ2

Skewness: μ1 – μ2 / (μ1 + μ2)3/2

## Uses of the Skellam distribution

Probability generating function (PGF): e-(μ1 + μ2) + μ1t + μ2/t

The Skellam distribution is a good choice for modeling data that is expected to be approximately Poisson distributed, such as the difference between the number of births and deaths in a population. This is the original use case for the Skellam distribution, and it is still a common use case today. Other use cases include:

• Image detection and denoising .
• Modeling the difference between the number of arrivals and departures at a transportation hub .
• Modeling noise in PET imaging .
• Showing the spread of points in sports, where points are equal (in other words, games such as hockey or soccer where goals always equal one point) .
• Studying treatment effects .
• Creating the Skellam mechanism for Differentially Private Federated Learning. Differential privacy is a technique that adds noise to datasets to protect individual privacy, while federated learning enables model training without sharing data. Differentially private federated learning combines these techniques to train machine learning models on sensitive data from multiple devices. This technique is well-suited to training models on data like medical or financial data while keeping individual data private.

## References

 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.

 Hwang, Y. et al. Statistical background subtraction based on the exact per-pixel distributions. MVA2007 IAPR Conference on Machine Vision Applications, May 16-18, 2007, Tokyo, JAPAN from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.2875&rep=rep1&type=pdf

 Hirakawa, K. et al. Wavelet-based Poisson Rate Estimation using the Skellam distribution, in C.A. Bouman et al. (Eds.). Proceedings of SPIE 7246 (2009).

 Liu, X. Models for excess demand in urban environments. Doctoral thesis.

 M. Yavuz and J. A. Fessler. Maximum likelihood emission image reconstruction for randoms-precorrected pet scans. In IEEE Nuclear Science Symposium Conference Record, pages 15/229–15/233, 2000

 Karlis, D. & Ntoufras, I. Bayesian modelling of football outcomes: using the Skellam’s distribution for the goal difference. IMA J. Manag. Math 20(2) (2008), 133-145.

 Karlis, D. & Ntzoufras, I. Bayesian analysis of the differences of count data, Stat. Med. 25 (2006), 185-1905.

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