The Skellam distribution

Distributions, distributions everywhere

There are a ton of distributions out there; SciPy alone has several hundred and that's nowhere near a complete set. I'm going to talk about one of the lesser known distributions, the Skellam distribution, and what it's useful for. My point is a simple one: it's not enough for data scientists to know the main distributions, they must be aware that other distributions exist and have real-world uses.

Overview of the Skellam distribution

It's easy to define the Skellam distribution: it's the difference between two Poisson distributions, or more formally, the difference between two Poisson distributed random variables. 

So we don't get lost in the math, here's a picture of a Skellam distribution.

If you really must know, here's how the PMF is defined mathematically:

\[ P(Z = k; \mu_1, \mu_2) = e^{-(\mu_1 + \mu_2)} \left(\frac{\mu_1}{\mu_2}\right)^{k/2} I_k(2\sqrt{\mu_1 \mu_2}) \] where \(I_k(x)\) is given by the modified Bessel function: \[ I_k(x) = \sum_{j=0}^{\infty} \frac{1}{j!(j+|k|)!} \left(\frac{x}{2}\right)^{2j+|k|} \]

this all looks very complicated, but by now (2025), it's easy to code up, here's the SciPy code to calculate the PMF:

probabilities = stats.skellam.pmf(k=k_values, mu1=mu1, mu2=mu2)

What use is it?

Here are just a few uses I found:

• Finance: modeling price changes between trades.
• Medicine: modeling the change in the number of beds in an ICU, epileptic seizure counts during drug trials, differences in reported AIDS cases, and so on.
• Sports: differences in home and away team football or hockey scores.
• Technology: modeling sensor noise in cameras, 

Where did it come from?

Skellam published the original paper on this distribution in 1946. There isn't a lot of background on why he did the work and, as far as I can tell, it wasn't related to World War II research work in any way. It's only really been discussed more widely once people discovered it's use for modeling sports scores. It's been available as an off-the-shelf distribution in SciPy for over a decade now.

As an analyst, what difference does this make to you?

I worked in a place where the data we analyzed wasn't normally distributed (which isn't uncommon, a lot of data sets aren't normally distributed), so it was important that everyone knew at least something about non-normal statistics. I interviewed job candidates for some senior positions and asked them how they would analyze some obviously non-normal data. Far too many of them suggested using methods only suitable for normally distributed data. Some candidates had Master's degrees in relevant areas and told me they had never been taught how to analyze non-normal data, and even worse, they never looked into it themselves. This was a major warning for us recruiting.

Let's imagine you're given a new data set in a new area and you want to model it. It's obviously not normal, so what do you do? In these cases, you need to have an understanding of what other distributions are out there and their general shape and properties. You should just be able to look at data and guess a number of distributions that could work. You don't need to have an encyclopedic knowledge of them all, you just need to know they exist and you should know how to use a few of them.