Poisson processes
A homogeneous Poisson process counts events that occur independently at a constant rate \(\lambda>0\).
Event counts
If \(N(t)\) is the number of events by time \(t\), then
\[P(N(t)=k)=e^{-\lambda t}\frac{(\lambda t)^k}{k!}.\]Therefore
\[E[N(t)]=\lambda t,\qquad \operatorname{Var}(N(t))=\lambda t.\]Small intervals
For small \(\Delta t\), the probability of one event is approximately \(\lambda\Delta t\), while the probability of two or more events is of smaller order.
Key idea. The Poisson process provides a basic mathematical model for randomly timed events occurring at a constant average rate.