What is a Poisson process?
A Poisson process with rate λ is a continuous-time stochastic process \(N(t)\) that counts the number of events that have occurred up to time \(t\). It is characterized by three fundamental properties:
- Independent increments: the number of events in disjoint time intervals are independent random variables. What happens in \([0,s]\) does not affect what happens in \([s,s+t]\).
- Stationary increments: \(N(s+t) - N(s)\) has the same distribution as \(N(t)\) for all \(s \geq 0\). The distribution of events depends only on the length of the interval, not on its position in time.
- Orderliness (no simultaneous events): the probability of two or more events occurring in a very small interval \([t, t+\Delta t]\) is \(o(\Delta t)\), i.e. negligible relative to \(\Delta t\).
As a consequence of these properties, the number of events in any interval of length \(t\) follows a Poisson distribution with mean \(\lambda t\):
\[ P(N(t) = k) = \frac{e^{-\lambda t} (\lambda t)^k}{k!}, \quad k = 0, 1, 2, \ldots \]
The variance is also \(\lambda t\), which means the relative variability \(\sqrt{\lambda t}/(\lambda t) = 1/\sqrt{\lambda t}\) decreases over time: the longer the horizon, the more predictable (in relative terms) the number of events.
The exponential distribution as inter-event time
Let \(T_1\) be the time to the first event. Its distribution is easy to derive using the connection to the Poisson process:
\[ P(T_1 > t) = P(N(t) = 0) = e^{-\lambda t} \]
Therefore, \(T_1 \sim \text{Exponential}(\lambda)\) with density function \(f(t) = \lambda e^{-\lambda t}\), mean \(1/\lambda\), and median \(\ln(2)/\lambda\).
By the independent- and stationary-increments property, all the times between consecutive events \(T_1, T_2, T_3, \ldots\) are independent and identically distributed as \(\text{Exponential}(\lambda)\). This property is equivalent to the process being Poisson.
The exponential distribution satisfies the famous memoryless property:
\[ P(T > s + t \mid T > s) = P(T > t) \]
In other words, the remaining time until the next event does not depend on how long you have already been waiting. This is the only continuous distribution with this property — the continuous analogue of the geometric distribution (the only discrete one with the memoryless property).
Applications of the Poisson process
The Poisson process is the reference model for rare events, or events that arrive randomly at a constant rate. Some classic applications:
- Queueing theory (M/M/1): customer arrivals at a service point (a bank, a counter, a web server) are frequently modeled with a Poisson process with rate λ. M/M/1 queues also assume that service times are exponential with rate μ. System stability requires ρ = λ/μ < 1.
- Radioactive decay: the emission of alpha or beta particles from a radioactive sample follows (with great precision) a Poisson process, since each nucleus decays independently with constant probability per unit of time.
- Network traffic: data packets on telecommunications networks, requests to a web server, or incoming phone calls are modeled with Poisson processes as a first approximation.
- Insurance claims: in actuarial science, the number of claims in a period is modeled with a Poisson distribution (a compound Poisson process when each claim has a random amount).
- Epidemiology of rare diseases: the incidence of rare diseases in a population of size n, with small probability p of contracting it, can be approximated by Poisson(np).
- Astronomy and particle physics: photons arriving at a detector, cosmic rays hitting a given area, or collisions in a particle accelerator.
Generalizations of the Poisson process
The homogeneous Poisson process (constant rate λ) has several important generalizations:
- Non-homogeneous Poisson process: the rate varies with time, \(\lambda(t)\). The expected number of events in \([0, T]\) is \(\int_0^T \lambda(t)\,dt\). Useful when the intensity of the process changes over the course of the day (e.g., phone calls: more frequent at noon than at 3 a.m.).
- Compound Poisson process: each event carries an associated random "size" or "severity" \(Y_i\). The process \(S(t) = \sum_{i=1}^{N(t)} Y_i\) is a compound Poisson process. It is the basic model in insurance risk for total claims.
- Cox process (doubly stochastic Poisson): the rate \(\lambda\) is itself a stochastic process. It captures the overdispersion observed when the empirical variance exceeds the mean (something the pure Poisson cannot model).
- Hawkes process: a self-exciting Poisson process in which each event temporarily raises the future intensity. It models contagion phenomena (e.g., earthquake aftershocks, the spread of news on social media, orders in financial markets).
- Marked Poisson process: a generalization in which each event carries a "mark" that can contain additional information (event type, spatial location, magnitude, etc.).
Frequently asked questions
Why does the rate λ appear in both the Poisson and the exponential distributions?
They are two sides of the same coin. λ is the occurrence rate of the process: how many events we expect per unit of time. In the Poisson distribution, λt is the expected number of events in the interval \([0,t]\). In the exponential distribution, \(1/\lambda\) is the expected time until the next event. If λ is large (many events per unit of time), the mean \(1/\lambda\) of the time between events is small — arrivals are frequent and the waits are short. Both formulations are equivalent: specifying one fully determines the other.
Can the Poisson process be used when events are not fully independent?
Not directly. Independence of increments is an essential property of the Poisson process: if knowledge of what has happened in the past changes our expectation of what will happen in the future (beyond the elapsed time), the process is not Poisson. For events with temporal correlation — e.g., earthquakes (an aftershock raises the probability of another), social media activity (a viral tweet generates more replies), or credit data (one customer's delinquency influences that of their neighbors) — richer models are used, such as Hawkes processes, Poisson processes with frailty, or generalized renewal processes.
How does the Poisson process relate to the binomial distribution?
When \(n \to \infty\) and \(p \to 0\) with \(np = \lambda\) fixed, the \(\text{Binomial}(n, p)\) distribution converges to the \(\text{Poisson}(\lambda)\) distribution. This is the law of rare events (or Poisson limit theorem). Intuitively: divide the interval \([0,T]\) into \(n\) small subintervals of length \(T/n\). An event can occur in each with probability \(p \approx \lambda T/n\). The total number of events is approximately \(\text{Binomial}(n, p)\), which converges to \(\text{Poisson}(\lambda T)\) as \(n \to \infty\). This derivation also justifies the properties of the Poisson process constructively.
How should I interpret it when the empirical variance matches the mean?
In the Poisson distribution, the theoretical variance equals the mean: \(\text{Var}(N(T)) = \mathbb{E}[N(T)] = \lambda T\). If in your simulations the empirical mean and variance are similar, that confirms the simulated process is indeed Poisson. If the empirical variance greatly exceeds the mean (overdispersion), it could indicate that the real underlying model is not pure Poisson, and it would be worth considering a negative binomial distribution or a Cox process.