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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 λ λ {\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 B {\displaystyle {\mathcal {B}}} . More precisely, the parameters are λ λ {\displaystyle \lambda } and a probability distribution on compact sets; for each point ξ ξ {\displaystyle \xi } of the Poisson point process we pick a set C ξ ξ {\displaystyle C_{\xi }} from the distribution, and then define B {\displaystyle {\mathcal {B}}} as the union ∪ ∪ ξ ξ ( ξ ξ + C ξ ξ ) {\displaystyle \cup _{\xi }(\xi +C_{\xi })} of translated sets.

To illustrate tractability with one simple formula, the mean density of B {\displaystyle {\mathcal {B}}} equals 1 − − exp ⁡ ⁡ ( − − λ λ A ) {\displaystyle 1-\exp(-\lambda A)} where Γ Γ {\displaystyle \Gamma } denotes the area of C ξ ξ {\displaystyle C_{\xi }} and A = E ⁡ ⁡ ( Γ Γ ) . {\displaystyle A=\operatorname {E} (\Gamma ).} The classical theory of stochastic geometry develops many further formulae. cite-ref-1[1]cite-ref-2[2]

As related topics, the case of constant-sized discs is the basic model of continuum percolationcite-ref-3[3] and the low-density Boolean models serve as a first-order approximations in the study of extremes in many models.cite-ref-4[4]

cite-note-11. ↑ citerefstoyan-d-kendall-w-s-mecke-j-1987Stoyan, D.; Kendall, W.S. & Mecke, J. (1987). Stochastic geometry and its applications. Wiley.

cite-note-22. ↑ citerefschneider-r-weil-w-2008Schneider, R. & Weil, W. (2008). Stochastic and Integral Geometry. Springer.

cite-note-33. ↑ citerefmeester-r-roy-r-2008Meester, R. & Roy, R. (2008). Continuum Percolation. Cambridge University Press.

cite-note-44. ↑ citerefaldous-d-1988Aldous, D. (1988). Probability Approximations via the Poisson Clumping Heuristic. Springer.