Poisson Processes

A Poisson process is the continuous-time version of the rare-event story from the Poisson distribution. It says that events arrive independently over disjoint time intervals, at a constant average rate $\lambda$. The same object can be described by exponential waiting times, Poisson counts, or infinitesimal Bernoulli trials.

1. Arrival timeline and the two distributions

If interarrival times are iid $T_i\sim\mathrm{Exp}(\lambda)$, then arrivals occur at $S_i=T_1+\cdots+T_i$. The count in a window obeys $N(t)\sim\mathrm{Poisson}(\lambda t)$. This first figure keeps both views visible at once.

2. Memorylessness

The exponential distribution is memoryless: $P(T>s+t\mid T>s)=P(T>t)$. If no event has arrived by time $s$, the additional wait has the same distribution as a fresh wait. This is the only continuous distribution with that property.

3. Three equivalent definitions

These are three ways to specify the same process:

1. Axiomatic. $N(t)\sim\mathrm{Poisson}(\lambda t)$ with stationary independent increments.
2. Infinitesimal. $P(N(h)=1)=\lambda h+o(h)$ and $P(N(h)>1)=o(h)$.
3. Time-centric. iid $T_i\sim\mathrm{Exp}(\lambda)$ and arrival times $S_i=\sum_{k\le i}T_k$.

4. Superposition, thinning, splitting

Poisson processes are stable under simple rate operations. Independent streams merge by adding rates. Keeping each event with probability $p$ thins the rate to $p\lambda$. Splitting events into categories creates independent Poisson streams with category rates.

5. Is your data Poisson?

A Poisson-process model needs more than Poisson-looking totals. The rate should scale with exposure length, and disjoint intervals should behave independently. The dispersion index $\mathrm{Var}(N)/\mathbb{E}N$ is a fast count diagnostic: it should be near 1 for equal windows.