Boolean model (probability theory)

For statistics in probability theory, 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

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

and a probability distribution on compact sets; for each point

of the Poisson point process we pick a set

To illustrate tractability with one simple formula, the mean density of

The classical theory of stochastic geometry develops many further formulae.

[1][2] As related topics, the case of constant-sized discs is the basic model of continuum percolation[3] and the low-density Boolean models serve as a first-order approximations in the study of extremes in many models.

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