Chapter 10 Poisson processes

The Poisson process is a model for counting the number of events that have occurred up to some time \(t\). It is a continuous-time stochastic process \(X=(X_t)_{t\in[0,\infty)}\) with \(X_0=0\) (starting our counter at zero events), and \(X_t\) is the number of events that have occurred since time 0 to time \(t\). There is a single parameter, \(\lambda\), and it is called by several different names: the rate parameter, the intensity, or the arrival rate. It is the mean number of events per unit time (or the expected number of arrivals in a unit time interval). We write \(X\sim\mathsf{Poi}_\lambda\) to denote \(X=(X_t)_{t\in[0,\infty)}\) is a Poisson process with arrival rate \(\lambda\).

The underlying assumptions are that non-overlapping time intervals are independent, and that the number of events on any time interval is a Poisson random variable with mean \(\lambda\) multiplied by the length of the interval. That is, \(X_t\sim\mathsf{Pois}(\text{mean}=\lambda t)\).

test edit