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 .  More precisely, the parameters are  and a probability distribution on compact sets; for each point  of the Poisson point process we pick a set  from the distribution, and then define  as the union 
 of translated sets.

To illustrate tractability with one simple formula, the mean density of  equals  where  denotes the area of  and   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