How many customers will arrive at a bank in the next hour? How many radioactive atoms will decay in the next second? How many typos appear per page in a long manuscript? These questions share a common structure: they ask about the count of independent, rare events in a fixed interval. The Poisson distribution models exactly this situation, arising naturally whenever events occur randomly in time or space at a constant average rate. Named for Siméon Denis Poisson, who introduced it in 1837, it is one of probability's most widely applicable models.
Deriving the Distribution
The Poisson distribution emerges as the limit of a binomial distribution. Consider n trials each with small probability p of success, with mean λ = np held constant as n → ∞. As n grows, the binomial probabilities converge to: P(X = k) = e^{-λ} λ^k / k!, for k = 0, 1, 2, …. Both the mean and variance of this distribution equal λ—a distinctive property that provides a quick check: if the mean and variance of count data are approximately equal, Poisson may be appropriate. If the variance substantially exceeds the mean (overdispersion), extensions like the negative binomial are needed.
Properties
Several elegant properties make the Poisson distribution mathematically tractable. Additivity: the sum of independent Poisson(λ₁) and Poisson(λ₂) random variables is Poisson(λ₁ + λ₂). Splitting: each event in a Poisson process independently classified as type A with probability p gives a Poisson(λp) count for type A events. These properties allow modeling situations where events arrive from multiple independent sources or where different event types are tracked separately. The moment generating function and probability generating function of the Poisson distribution have particularly clean forms, making theoretical calculations tractable.
The Poisson Process
The Poisson distribution is the marginal distribution of the Poisson process: a continuous-time model where events occur randomly at constant rate λ. The number of events in any time interval [0,t] follows Poisson(λt). Intervals between successive events follow an exponential distribution with mean 1/λ. The process has no memory: the time since the last event contains no information about when the next will occur. This memoryless property makes the Poisson process the continuous-time analog of Bernoulli trials and the simplest model for queues, radioactive decay, and financial order flow.
Applications
The Poisson distribution appears across science and engineering. Radioactive decay: the number of atoms decaying per second in a sample follows a Poisson distribution—this was one of the original applications. Queueing theory: arrivals to a service system (customers, phone calls, web requests) are modeled as Poisson processes, enabling analysis of wait times and server utilization. Epidemiology: rare disease incidence rates in a population follow approximately Poisson distributions. Ecology: the number of organisms found in randomly chosen sampling plots often fits Poisson, providing a null model for spatial distribution analysis.
When Poisson Fails
The Poisson model requires independence and constant rate—assumptions that fail in many real contexts. Earthquake aftershocks cluster in time (positive dependence). Website traffic spikes around news events (non-constant rate). Overdispersion (variance > mean) in count data suggests clustering or unobserved heterogeneity in rates. Underdispersion (variance < mean) suggests inhibition or regularity. The negative binomial distribution handles overdispersion by allowing the rate λ to vary across observations. Zero-inflated models address excess zeros in count data, common in medical applications where many subjects experience zero events.
Conclusion
The Poisson distribution is the natural model for counting rare, independent events—a role it fills across physics, biology, engineering, and business. Its derivation as a binomial limit, its elegant mean-equals-variance property, and its connection to the Poisson process make it both mathematically beautiful and practically versatile. Understanding when Poisson applies—and recognizing its failures—is fundamental statistical literacy for anyone working with count data, which arises in virtually every scientific and technical domain.