The Poisson distribution models the occurrence of rare, independent events across continuous intervals of time, space, or volume.
It is derived as the mathematical limit of the binomial distribution as trial count approaches infinity while success probability approaches zero.
The distribution is defined by a single positive parameter, lambda, which represents the expected event rate across the interval.
A unique mathematical property of the Poisson distribution is equidispersion: its theoretical mean and variance are both equal to lambda.
In empirical data, observed variance frequently exceeds the mean, a condition known as overdispersion.
Overdispersion indicates that events may be clustered, rates may fluctuate over time, or the independence assumption is violated.
When overdispersion occurs, data scientists often replace the Poisson model with a negative binomial distribution.
The Poisson formula calculates the probability of observing exactly k events using powers of lambda, factorials, and Euler constant e.
The time elapsed between consecutive Poisson events follows a continuous exponential distribution with rate parameter lambda.
This memoryless connection between Poisson arrivals and exponential waiting times forms the foundation of modern queuing theory.
Applications include modelling website traffic surges, call centre arrival spikes, earthquake frequencies, and insurance accident claims.
In retail management, Poisson models optimise inventory stocking levels for slow-moving products with irregular sales patterns.
Poisson regression extends the model to predict event counts conditioned on multiple explanatory feature variables.
A key limitation is that Poisson models cannot accommodate simultaneous events occurring at the exact same instant.
A common error is applying Poisson models to count data where events are mutually repulsive or strictly scheduled in advance.