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For statistics in probability theory, the: Boolean-Poisson model /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 ${\displaystyle \lambda }$ 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 ${\displaystyle {\mathcal {B}}}$. More precisely, the parameters are ${\displaystyle \lambda }$ and a probability distribution on compact sets; for each point ${\displaystyle \xi }$ of the Poisson point process we pick a set ${\displaystyle C_{\xi }}$ from the distribution. And then define ${\displaystyle {\mathcal {B}}}$ as the union ${\displaystyle \cup _{\xi }(\xi +C_{\xi })}$ of translated sets.

To illustrate tractability with one simple formula, the mean density of ${\displaystyle {\mathcal {B}}}$ equals ${\displaystyle 1-\exp(-\lambda A)}$ where ${\displaystyle \Gamma }$ denotes the area of ${\displaystyle C_{\xi }}$ and ${\displaystyle A=\operatorname {E} (\Gamma ).}$ 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.

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