Poisson Distribution

Math for CS · Axiom Academy

Modeling rare events and arrival rates The Poisson distribution models the number of events occurring in a fixed interval of time or space, when events happen at a constant average rate and independently of each other. Events occur at a constant average rate Events are independent (one doesn't trigger another) Two events can't happen at the exact same instant Examples: server requests per minute, typos per page, cosmic ray hits on a chip per hour, 911 calls per day in a city. Parameter: (average rate / expected count) The Poisson distribution arises as a limit of the Binomial when: Packet arrivals on a network link are often modeled as a Poisson process. This underpins queuing theory (M/M/1 queues), which determines buffer sizes and expected wait times. In distributed databases, write conflicts in a time window follow approximately Poisson statistics when many independent clients write at low individual rates. Cache misses under random access patterns. If each access has a small probability of being a miss, total misses per interval are roughly Poisson. Intrusion attempts, brute-force login events, and DDoS packet arrivals are modeled as Poisson processes to design rate limiters and alert thresholds. Application: If two independent servers handle and requests per second, a combined monitor sees events per second. Poisson counts events in a fixed interval at constant rate Limit of Binomial (n,p) as , , Independent Poisson variables add:

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