Picture a sample space \(\Omega\) (e.g. \(\mathbb{R}\)), with some intensity function \(f: \Omega \rightarrow \mathbb{R}_{\geq 0}\). For a given interval \(A \subset \Omega\), the random number of points that fall into the interval A is distributed Poisson with parameter equal to the integral of the intensity over the region:
\[N_A \sim Poisson(\int_{a \in A} f(a) da )\]with the additional property that if two regions \(A, B\) are disjoint, then \(N_A\) and \(N_B\) are disjoint. This integral \(\int_A f(a) da\) can be viewed as a measure, which presents an equivalent view, that \(N(\cdot) \sim Poisson(F(\cdot))\) is a random measure.
A stochastic process \({N_t : t \geq 0}\) is said to be a Poisson point process if the process has three properties:
TODO
An alternative, equivalent definition is that the number of points on the real line in the interval \((a, b]\) is distributed Poisson with rate \((b-a) \lambda\) i.e.
\[P(N(a, b] = n) = \frac{(\lambda(b-a))^n e^{-\lambda(b-a)}}{n!}\]