Poisson distributions

Let’s take a brief look at one alternative distribution since it will come up later: the Poisson distribution, which models the number of discrete events seen in an interval of time or space; for example, radioactive decay. It’s given by:

(5)\[P(k;\lambda) = \frac{\lambda^{k}e^{- \lambda}}{k!}\]

where:

  • \(k\) is the number of occurrences you might measure in an interval,

  • \(\lambda\) is the mean number of events you’d expect see per interval,

  • \(P(k;\lambda)\) is the probability of seeing \(k\) events given a mean of \(\lambda\).

This plot may help you visualize what’s going on:

poisson-distributions

Figure 91: Examples of Poisson distributions for a few values of \(\lambda\). Source: Skbkekas under the CC BY 3.0 license

For example, if you’re running an experiment whose observations follow a Poisson distribution, and the mean number of events you’d expect to see is \(\lambda\)=4, then the probability of seeing exactly 4 events in the interval is about 0.20.

You may have noticed that for \(\lambda\)=10, the curve looks like a Gaussian. In fact, a Gaussian can be defined as the limit for large \(\lambda\) of a Poisson distribution.1

As it happens, the particle-physics experiments that I’ve worked on have typically involved tens to hundreds of thousands of events, even after applying cuts,2 so I’ve usually worked with Gaussian probability distributions. If you’re working on an experiment with infrequent events in space or time, then you’ll become more familiar with the Poisson distribution.

Let’s keep that in mind as we move on to discussing fitting a function to data in the next section.

xkcd poisson

Figure 92: https://xkcd.com/12/ by Randall Munroe. This early xkcd cartoon is more subtle than usual. If you’re having trouble figuring it out, take another look at the above equation and plot.

1

Why is it a “Gaussian” distribution (making an adjective from the name Carl Friedrich Gauss) but a “Poisson” distribution (after Siméon-Denis Poisson) and not a “Poissonian” distribution? I’m a physicist, not a language expert. I’ll leave the answer as an exercise for the student.

2

If you don’t know what “applying a cut” means yet, you can read the RDataFrame explanation, the Python explanation, or the C++ explanation.