A point process is a random scattering of point in a given space over time. Poisson processes are a fundamental type of point process that are used in a variety of scenario in probability and/or statistics. For instance we want to model the total number of clients \(N(t)\) that have visited a shop till a certain time \(t\). Poisson process can be used to develop probabilistic models for \(N(t\)), where, in this case, the arrival times of the clients form a Poisson point process. The Poisson process has two key properties: the Poisson property (Poisson distribution of points count) and the independence property, where the former implies the latter. Let's see what they are about.
Poisson distribution of point counts
A Poisson point process is characterized through the Poisson distribution, that is the statistical distribution [1]
of a random variable \(X\). The latter is a Poisson random variable with mean \(\mu\) if it has probability mass function given by:
\[(X = n) = \frac{ e^{-\mu} \mu ^n}{n!} \ \text{ for $n = 0,1, 2, \ldots$}\]
where \(n!\) is the factorial and the \(\mu^n\) determines the shape of the distribution.
By this definition, a Poisson point process is characterized by the unique property that the count of points within a defined region of the
process's underlying space follows a random variable with a Poisson distribution.
Independence property
Imagine a set of separate and well-defined subregions within the underlying space. By definition, in a Poisson point process, the number of
points within each distinct subregion is entirely independent and unrelated to the
counts in all other subregions. This property is also known as complete randomness or complete independence.
In essence, there is absence of interation between different regions and the points themselves.