PoissonProcess

A Poisson process is a mathematical model used to describe the occurrence of events over time or space. It is characterized by the assumption that events occur independently and randomly, with a constant average rate. The Poisson process is commonly used to model random events.

There are a few key properties that define a Poisson process:

• The number of events in non-overlapping intervals is independent.
• The probability of an event occurring in a short interval of length $\Delta t$ is proportional to the length of the interval, i.e., $\lambda \Delta t$, where $\lambda$ is the average rate of events.
• The probability of more than one event occurring in a short interval of length $\Delta t$ is negligible as $\Delta t$ approaches zero.

The Poisson process is completely determined by its rate parameter $\lambda$, which represents the average number of events per unit of time or space. The number of events occurring in an interval of length $t$ follows a Poisson distribution with parameter $\lambda t$:

$$P(N(t) = n) = \frac{(\lambda t)^n e^{-\lambda t}}{n!}$$

where $N(t)$ is the number of events in the interval of length $t$, and $n$ is a non-negative integer.

In summary, a Poisson process is a simple yet powerful model for describing the random and independent occurrence of events over time or space with a constant average rate.