Poisson point process

Definition

Let $\lambda > 0$. A homogeneous Poisson process with rate $\lambda$ is a stochastic process $(N(t))_{t \ge 0}$ with values in $\{0, 1, 2, \dots\}$ such that:

1. $N(0) = 0$.
2. For all $0 \le t_0 < t_1 < \dots < t_n$, the increments $N(t_1) - N(t_0), \dots, N(t_n) - N(t_{n-1})$ are independent.
3. For all $0 \le s < t$, $N(t) - N(s)$ has the Poisson distribution with parameter $\lambda (t - s)$.

Remarks

$N(t)$ counts the points, or events, in the interval $(0, t]$. One usually also asks that the paths $t \mapsto N(t)$ are right-continuous. The times between consecutive events are i.i.d. with the exponential distribution $\operatorname{Exp}(\lambda)$. Conversely, i.i.d. $\operatorname{Exp}(\lambda)$ gaps define a Poisson process with rate $\lambda$. Given $N(t) = n$, the $n$ event times in $(0, t]$ have the law of $n$ independent uniform points in $(0, t]$, put in increasing order. On a general space $E$ with a measure $\mu$, a Poisson point process has independent counts in disjoint sets. The count in a set $B$ has the law $\operatorname{Poisson}(\mu(B))$ when $\mu(B) < \infty$.

Explanations

L1 human

The Poisson process models events that happen "completely at random" in time, such as clicks of a Geiger counter or arrivals of customers. On average, $\lambda$ events happen per unit of time, and the waiting times between events are exponential.