Poisson Processes

Poisson processes model the number of randomly occurring events in a fixed time or space interval, with a constant average rate. They represent a special case of continuous-time Markov chains where we focus on counting events rather than tracking system states.

A Poisson process N(t) must satisfy several key properties: N(0) = 0 (starting at zero), it has independent increments (events in non-overlapping intervals are independent), and for small intervals Δt, the probability of exactly one event is approximately λΔt while the probability of multiple events is negligible.

The number of events N(t) in an interval of length t follows a Poisson distribution with parameter λt: P(N(t) = k) = e^(-λt)(λt)^k/k!, where λ is the rate parameter. The time between consecutive events follows an exponential distribution with mean 1/λ.

These processes are applied in machine learning for modeling customer arrivals, network traffic, failure events, and any situation where discrete events occur randomly over time or space with known average rates.

In recommendation systems, Poisson processes help model user interaction patterns. In anomaly detection, deviations from expected Poisson behavior can signal unusual activity. Healthcare applications include modeling patient arrivals at emergency rooms, disease outbreaks, and mutation occurrences in genetic sequences.