1. Poisson processes and the Poisson distribution
In his great book on information theory, statistical inference, and machine learning, David MacKay described a town called Poissonville where the buses have a very erratic schedule.
If you stand at a bus stop in Poissonville, the amount of time you have to wait for a bus is completely independent of when the previous bus arrived.
In other words, you would watch a bus drive off and another one will arrive almost instantly, or you could be waiting for hours.
Arrival of buses in Poissonville is what we call
2. Poisson process
a Poisson process. The timing of the next event is completely independent of when the previous event happened. Many real-life processes behave in this way.
3. Examples of Poisson processes
For example, natural births in a given hospital are a Poisson process. There is a well-defined average number of natural births per year, and the timing of one birth is independent of the timing of the previous one.
Another example is hits on a website. The timing of the next hit is independent of the timing of the last.
There are countless other examples. Any process that matches the buses in Poissonville story is a Poisson process. The number of arrivals of a Poisson process in a given amount of time is Poisson distributed.
4. Poisson distribution
The Poisson distribution has one parameter, the average number of arrivals in a given length of time.
So, to match the story, we could consider the number of hits on a website in an hour with an average of six hits per hour. This is Poisson distributed.
5. Poisson PMF
Let's take a look at the PMF for this example. For a given hour, we are most likely to get 6 hits, which is the average, but we may get more than ten, or possibly none. You might notice that this looks an awful lot like the Binomial PMF we looked at before. This is because the Poisson distribution
6. Poisson Distribution
is a limit of the Binomial distribution for low probability of success and large number of trials, i.e., for rare events. You'll explore this relationship further in the following interactive exercises.
7. The Poisson CDF
To sample from the Poisson distribution, we use rng-dot-poisson. It also has the size keyword argument to allow multiple samples. Let's use this function to generate the Poisson CDF.
After that it is the usual procedure of computing the ECDF, plotting it, and labeling axes.
8. The Poisson CDF
The result, predictably, looks like the Binomial CDF we saw before.
9. Let's practice!
Now let's get some experience with the Poisson distribution in the exercises!