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
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
. More precisely, the parameters are
λ
and a probability distribution on compact sets; for each point
ξ
of the Poisson point process we pick a set
C
ξ
from the distribution, and then define
B
as the union
∪
ξ
(
ξ
+
C
ξ
)
of translated sets.

To illustrate tractability with one simple formula, the mean density of
B
equals
1
−
exp
(
−
λ
A
)
where
Γ
denotes the area of
C
ξ
and
A
=
E
(
Γ
)
.
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.