---
tag: 2J9GB49FHA
kind: concept
name: Stochastic geometry
msc: [60D05, 60G55, 60G57]
sources:
  - url: https://en.wikipedia.org/wiki/Stochastic_geometry
same_as:
  wikidata: Q1666739
  wikipedia: Stochastic geometry
---

## Summary

Stochastic geometry is the part of probability theory that studies random geometric structures in space.
Its main objects are [[point processes|DNDKBVHS3H]], random closed sets, random tessellations and random measures.
Typical models are the [[Poisson point process|0MTNR28K60]] and the [[Boolean model|NK3CZTVEJD]].
It supplies models and formulas for random patterns in materials science, wireless networks, biology and image analysis.

## Explanations

### L1 human

Stochastic geometry asks questions about shapes that are placed at random.
Think of raindrops on a pavement, trees in a forest, or cells seen under a microscope.
One models the centres as random points and the objects as random shapes around them.
For example, drop discs of radius $r$ at the points of a Poisson process of intensity $\lambda$ in the plane.
Then the fraction of the plane covered by the discs is $1 - e^{-\lambda \pi r^2}$.
Classical questions of geometric probability, such as [[Buffon's needle problem|06X5PF6YFP]], belong to the same circle of ideas.