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 <math>\lambda</math> 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 <math>{\mathcal B}</math>. More precisely, the parameters are <math>\lambda</math> and a probability distribution on compact sets; for each point <math>\xi</math> of the Poisson point process we pick a set <math>C_\xi</math> from the distribution, and then define <math>{\mathcal B}</math> as the union
<math>\cup_\xi (\xi + C_\xi)</math> of translated sets.
To illustrate tractability with one simple formula, the mean density of <math>{\mathcal B}</math> equals <math>1 - \exp(- \lambda A)</math> where <math>\Gamma</math> denotes the area of <math>C_\xi</math> and <math>A=\operatorname{E} (\Gamma).</math> The classical theory of stochastic geometry develops many further formulae.
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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.[4]
