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Poisson Process

A continuous-time counting process tracking the number of random events that have occurred by time $t$, where events happen at a constant average rate, independently of one another, and the gaps between them are exponentially distributed.

Stochastic ProcessesIntermediateFree

In Plain English

The Poisson process is the standard model for events trickling in randomly over time at some steady average rate, customer arrivals at a store, radioactive decay clicks on a Geiger counter, website hits per minute. Nothing about when the last event happened changes the odds of when the next one arrives, arrivals are memoryless, which is exactly why the gaps between consecutive events follow the exponential distribution. This entry treats the process through its temporal, counting-process lens, and as the canonical example of a continuous-time Markov chain (specifically a pure-birth process); see Poisson Point Process for the same underlying mathematics applied to points scattered in space instead of events unfolding in time.

Definition

A Poisson process with rate $λ > 0$ is a counting process ${N (t)}_{t \geq 0}$ , $N (t)$ giving the number of events that have occurred by time $t$ , satisfying: $N (0) = 0$ ; independent increments (the number of events in disjoint time intervals are independent); and $N (t + s) - N (t) \sim Poisson (λ s)$ for all $t, s \geq 0$ (stationary, Poisson-distributed increments). Equivalently, the process can be constructed via i.i.d. exponentially distributed interarrival times $T_{i} \sim Exp (λ)$ , a consequence of the exponential distribution being the unique continuous memoryless distribution. The Poisson process is simultaneously a Levy process (stationary, independent increments), a continuous-time Markov chain (specifically, a pure-birth process with constant birth rate $λ$ and no deaths), and the temporal building block for the compound Poisson process and Cox process. It is also the canonical reference model against which departures, clustering (as in a Hawkes process) or under-dispersion, are measured in event-timing data.

Formula

P (N (t) = k) = \frac{( λ t ) ^{k} e ^{- λ t}}{k !}, T_{i} \sim Exp (λ) i.i.d.

Notation

N (t)

—Number of events that have occurred by time $t$

λ

—Rate parameter: average number of events per unit time

T_{i}

—The $i$-th interarrival time, exponentially distributed with rate $\lambda$

Properties

The Poisson process's memorylessness (via the exponential interarrival distribution) is often the very first assumption checked, and often violated, in real event data: bursty, clustered arrivals (customer complaints after an outage, aftershocks after an earthquake) call for extensions like the Hawkes process instead.
As the unique pure-birth continuous-time Markov chain with constant rate, the Poisson process sits at the foundation of an entire hierarchy of more elaborate models, allow the rate to be itself random and you get a Cox process, let event sizes vary and you get a compound Poisson process, let past events boost future rates and you get a Hawkes process.
Despite sharing deep mathematical roots (both are built from the exponential distribution's memorylessness), the Poisson process's temporal framing, in queueing theory, reliability engineering, and actuarial claim-count modeling, is applied quite differently in practice from the spatial framing used for point pattern analysis in ecology and epidemiology.

At a Glance

Counts

N (t) \sim Poisson (λ t)

: number of events by time

t

Interarrival timesi.i.d. Exponential(

λ

), a consequence of memorylessness

As a Markov processa pure-birth continuous-time Markov chain (constant birth rate, no deaths)

Foundational forcompound Poisson, Cox, and Hawkes processes all build on it

Last updated September 8, 2026← Back to the Dictionary