Poisson point process
Spatial Poisson point process

A spatial Poisson process is a Poisson point process defined in the plane R 2 {\displaystyle \textstyle \mathbb {R} ^{2}}. For its mathematical definition, one first considers a bounded, open or closed (or more precisely, Borel measurable) region B {\textstyle B} of the plane. The number of points of a point process N {\displaystyle \textstyle N} existing in this region B ⊂ R 2 {\displaystyle \textstyle B\subset \mathbb {R} ^{2}} is a random variable, denoted by N (B) {\displaystyle \textstyle N(B)}. If the points belong to a homogeneous Poisson process with parameter λ > 0 {\displaystyle \textstyle \lambda >0}, then the probability of n {\displaystyle \textstyle n} points existing in B {\displaystyle \textstyle B} is given by:

where | B | {\displaystyle \textstyle |B|} denotes the area of B {\displaystyle \textstyle B}.

For some finite integer k ≥ 1 {\displaystyle \textstyle k\geq 1}, we can give the finite-dimensional distribution of the homogeneous Poisson point process by first considering a collection of disjoint, bounded Borel (measurable) sets B 1, …, B k {\displaystyle \textstyle B_{1},\dots,B_{k}}. The number of points of the point process N {\displaystyle \textstyle N} existing in B i {\displaystyle \textstyle B_{i}} can be written as N (B i) {\displaystyle \textstyle N(B_{i})}. Then the homogeneous Poisson point process with parameter λ > 0 {\displaystyle \textstyle \lambda >0} has the finite-dimensional distribution:
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