Poisson process

A counting process in which events occur independently at a constant average rate, so counts in disjoint time intervals are independent Poisson random variables.

Poisson distribution: A discrete probability distribution for nonnegative integer counts, determined by a single rate parameter. The number of events in an interval of length t has this distribution with mean rate times t.
Stochastic process: A collection of random variables indexed by time, space, or another parameter. A Poisson process is one specific model of random change over time.
Queueing theory: The mathematical study of waiting lines, service systems, and congestion. Poisson arrivals often model customers or jobs entering a queue.
Nonhomogeneous Poisson process: A Poisson counting process whose event rate varies with time. It relaxes the constant-rate assumption while retaining independent increments.
Exponential distribution: A continuous probability distribution commonly used to model waiting times between events occurring at a constant rate. Successive waiting times between events follow this distribution.
Poisson random measure: A random measure whose counts on disjoint sets are independent Poisson random variables. It generalizes event counts from time intervals to measurable regions.
Radioactive decay: The random transformation of unstable atomic nuclei into more stable nuclei, releasing radiation. Independent nuclear decay events are commonly modeled as a Poisson process over fixed populations.
Compound Poisson process: A process formed by summing random event magnitudes arriving according to a Poisson process. It tracks accumulated event sizes, rather than merely counting arrivals.
Independent increments: A property of a stochastic process in which changes over disjoint intervals are independent random variables. This independence is a defining condition for the process.
Point process: A random collection of points in a space, often used to model event locations or times. A Poisson process is the time-indexed point-process case with constant intensity.

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