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statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
in
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, the Boolean-Poisson model or simply Boolean model for a random subset of the plane (or higher dimensions, analogously) is one of the simplest and most tractable models in stochastic geometry. Take a Poisson point process of rate \lambda in the plane and make each point be the center of a random set; the resulting union of overlapping sets is a realization of the Boolean model . More precisely, the parameters are \lambda and a
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
on compact sets; for each point \xi of the Poisson point process we pick a set C_\xi from the distribution, and then define as the union \cup_\xi (\xi + C_\xi) of translated sets. To illustrate tractability with one simple formula, the mean density of equals 1 - \exp(- \lambda A) where \Gamma denotes the area of C_\xi and A=\operatorname (\Gamma). The classical theory of stochastic geometry develops many further formulae. As related topics, the case of constant-sized discs is the basic model of continuum percolation and the low-density Boolean models serve as a first-order approximations in the study of extremes in many models.


References

Spatial processes {{probability-stub