Poisson point process

A random collection of points in a space whose counts in disjoint regions are independent Poisson random variables, with means determined by an intensity measure.

Poisson distribution: A discrete probability distribution for nonnegative integer counts, with probability mass determined by one rate parameter. Counts in each finite-measure region follow this distribution, with parameter equal to its intensity.
Point process: A random countable collection of points, often represented by the number of points in each region. A Poisson point process is a particularly tractable member of this broader family.
Homogeneous Poisson point process: A Poisson point process whose intensity is constant across space. It models uniform random placement with a single rate per unit volume.
Binomial point process: A point process with a fixed number of independently sampled points in a region. Its total count is fixed, unlike the Poisson-distributed count in a finite region.
Intensity measure: A measure that gives the expected number of points in each measurable region. It specifies the mean count and spatial distribution of points.
Measure theory: The mathematical study of measures, which assign sizes to sets and support integration and probability. Intensity measures use its language to assign expected counts to arbitrary measurable regions.
Inhomogeneous Poisson point process: A Poisson point process whose intensity varies with location. It represents spatial patterns whose expected density changes across the domain.
Cox process: A point process that is Poisson conditional on a random intensity measure. Random intensity introduces extra variation and dependence absent under a fixed intensity measure.
Independent increments: A property in which random changes over disjoint intervals or regions are independent. Counts in disjoint regions are independent, the central structural condition of the process.
Random measure: A random object that assigns a nonnegative measure to sets, with measurable random values. Counting points in regions turns a point process into a random measure.

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Le Cam's theoremNarrower topic: Poisson approximations for event counts underpin broader models of random points.