STOCHASTIC PROCESSES 14
The Poisson Process
A Poisson process counts events that occur independently through continuous time at a constant rate.
The counting process
Let \(N(t)\) be the number of events observed by time \(t\), with
\[N(0)=0.\]For a homogeneous Poisson process with rate \(\lambda>0\),
\[\boxed{N(t)\sim\operatorname{Poisson}(\lambda t)}\]
1. Distribution
\[P(N(t)=k)=e^{-\lambda t}\frac{(\lambda t)^k}{k!},\qquad k=0,1,2,\ldots\]2. Mean and variance
\[E[N(t)]=\lambda t,\qquad \operatorname{Var}(N(t))=\lambda t.\]3. Independent increments
Numbers of events in non-overlapping time intervals are independent.
4. Stationary increments
The distribution of the number of events in an interval depends only on the interval length, not its location in time.
5. Small-time behaviour
For small \(h\),
\[P(\text{one event in }h)=\lambda h+o(h),\] \[P(\text{no event in }h)=1-\lambda h+o(h).\]6. Waiting times
The times between consecutive Poisson events are independent exponential random variables:
\[\tau_k\sim\operatorname{Exp}(\lambda).\]7. Connection between counting and waiting
\[\boxed{\text{Poisson event counts}\longleftrightarrow\text{exponential inter-event times}}\]
This connection is fundamental in continuous-time Markov modelling.