Poisson point process

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In probability theory, a Poisson point process is a particular kind of random process by which a set of isolated points are scattered about a line or a plane or a three-dimensional space or any of various other sorts of spaces. The concept is named after the French mathematician Siméon Denis Poisson. Often the term Poisson process is used to mean a Poisson point process in which the space in which isolated points are randomly scattered is a line, which in many applications represents time.

The Poisson point process is characterized by the following properties:

  • The numbers of isolated points falling within two regions A and B are independent random variables if A and B do not intersect each other;
  • The expected number of isolated points falling within a region A is the measure of the region A. This "measure" is often proportional to the area or volume of A, but sometimes more elaborate measures are used. But the measure must be defined in such a way that the measure of the union of regions that do not intersect each other is simply the sum of their measures.

A consequence of these characterizing properties is that the probability distribution of the number X of isolated points falling within any region A is a Poisson distribution, which means that

 \Pr(X=x) = \frac{(\mu(A))^x e^{-\mu(A)}}{x!}

where μ(A) is the measure of the region A.

Further reading[edit]