information:filtering
Interpreted as a counting process

The homogeneous Poisson point process, when considered on the positive half-line, can be defined as a counting process, a type of stochastic process, which can be denoted as { N (t), t ≥ 0 } {\textstyle \{N(t),t\geq 0\}}. A counting process represents the total number of occurrences or events that have happened up to and including time t {\textstyle t}. A counting process is a homogeneous Poisson counting process with rate λ > 0 {\textstyle \lambda >0} if it has the following three properties:

The last property implies:

In other words, the probability of the random variable N (t) {\textstyle N(t)} being equal to n {\textstyle n} is given by:

The Poisson counting process can also be defined by stating that the time differences between events of the counting process are exponential variables with mean 1 / λ {\textstyle 1/\lambda }. The time differences between the events or arrivals are known as interarrival or interoccurence times.