Poisson Processes

Independent arrivals in continuous time, modeled by counts, waiting times, and rate operations.

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.

Figure 1 · Timeline, interarrival histogram, and count histogram

events exponential fit Poisson fit

rate lambda 2

window T 5

replicates 400

Process gallery · Five fresh Poisson sample paths

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.

Figure 2 · Residual wait after surviving s

rate lambda 2

already waited s 1

3. Three equivalent definitions

These are three ways to specify the same process:

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

Figure 3 · One realization, three descriptions

rate lambda 2

grid h 0.2

Figure 3b · Nonhomogeneous Poisson process with draggable rate

$\lambda(t)$ events cumulative intensity $\Lambda(t)$

early rate 1.0

middle rate 5.0

late rate 2.0

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.

Figure 4 · Rate operations on event streams

lambda 1 1.5

lambda 2 2

keep p 0.45

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.

Figure 5 · Interarrival Q-Q and count dispersion

scenario Poisson

mean rate 2

Figure 6 · Conditional uniformity of event times given the count

ordered arrival times uniform order-statistic reference

condition on $N(T)=n$ 7

replicates 300

Connection back to distributions. A Poisson process is the process-level story behind Poisson counts and exponential waiting times. The distributions are the one-window shadows of the process.

What next

Distributions

Named Distributions

Review the Poisson count distribution and the exponential waiting-time distribution.

Processes

Markov Chains

Poisson processes are continuous-time counting processes; Markov chains are the discrete-state companion.

Random processes

Power Spectral Density

Another process-level view: what stationary randomness looks like in frequency.