 Poisson point process

A Poisson point process is a point process with independent Poisson distribution counts on disjoint measurable sets. Its intensity measure $\mu$ specifies those counts: for $\mu (A)<\infty$,Thus both the expected value and variance of $N(A)$ are $\mu (A)$. The independence requirement concerns disjoint test sets; overlapping sets can share points and have dependent counts.^[ref]

For a homogeneous Poisson point process in Euclidean space, $\mu (A)=\lambda |A|$, with constant intensity $\lambda$ and volume $|A|$. For example, intensity $2$ on the real line gives expected count $2$ and empty-interval probability $e^{-2}$ in a unit interval. Counts in two disjoint unit intervals are independent.

The counting measure formulation records multiplicity. A non-atomic intensity measure gives a simple point process, with no coincident points; an atom of a measure can instead carry a random multiplicity. Poisson random measures make that counting formulation explicit, while Poisson thinning theorem and the Superposition theorem for Poisson point processes explain useful ways of constructing related processes.