Poisson Distribution

I recently finished the excellent book by Ian G. on ‘How to Win the Premier League’. For anyone interested in sport (football), statistics and decision-making, it is a must read. I cant imagine what drew me towards the book?!

One section in it particularly interested me, was around the ‘Poisson distribution’. It is also known as the ‘Law of Small Numbers’. In simple terms, it describes how often certain events happen over a specific period of time or space, especially when those events are rare or occur randomly.

Imagine you run a small coffee shop. You want to know how likely it is that a certain number of customers will come in during an hour, but you can't predict exactly when they’ll come. If customers arrive randomly (without any pattern) and their average rate of arrival is known, the Poisson distribution can help you estimate things like:

"How likely is it that exactly 3 customers will come in the next hour?"
"How often will I get 5 or more customers in a single hour?"

It helps you quantify and put a percentage on the likelihood of an event happening.

It works well for situations where:

1. Events happen independently of each other (like customers arriving one after another).
2. You know the average number of events that happen in a fixed period (e.g., 10 customers per hour).
3. The events happen randomly.

In the book, Graham states that “People have a psychological blind spot when it comes to the Poisson distribution. They grossly underestimate the natural variability in outcomes. For example, if a team averages 1.5 goals per game playing at home against average opposition, the Poisson distribution predicts there will be a 22% chance of them scoring exactly zero goals against average opposition. If the team scores zero goals, it is tempting to assume they are a bad team but there was an appreciable chance (22%) that they would score zero. It is usually the case of mistaking luck for performance or noise for signal. When information is limited, we should be wary of drawing conclusions from observations.”

This is also an important concept within investment and can help when thinking about the likelihood of rare events occurring (extreme market movements, defaults, etc.). It can help by improving how we view asset allocation by ensuring we have a balance between risk and reward and tail-risk protection strategies. On the behavioural side, when infrequent market events happen, it is easy to over-react and create a narrative that the event was either possible to predict or will continue indefinitely. If we can understand that these uncommon conditions happen and are even statistically likely, it can help frame the experience better for us as investors.

If you say Poisson distribution three times, Christian Poulsen appears and you scare all the Liverpool fans away.