The Poisson distribution gives the probability of observing exactly k events in a fixed interval when events occur independently at a constant average rate. The single parameter lambda (the Greek letter) sets both the mean and the variance of the distribution: low lambda produces a distribution skewed toward zero; higher lambda shifts and widens the histogram until it begins to approximate a normal bell.\n\nThe diagram plots each integer outcome on the x-axis with a bar whose height equals the probability of that exact count. Common real-world examples: goals scored per football match (lambda around 2.7 for many leagues), emergency calls per hour, radioactive decay events per minute, typographical errors per page.
The probability mass function is P(k) = (lambda^k * e^(-lambda)) / k!, where k is 0, 1, 2, ... and lambda > 0.\n\nKey properties: the mean equals lambda; the variance equals lambda; the standard deviation is sqrt(lambda). Because mean equals variance, data that are more dispersed than Poisson (variance > mean) are called overdispersed and require a negative binomial model instead.\n\nFor large lambda the Poisson PMF converges to a normal distribution with mean lambda and standard deviation sqrt(lambda). A common rule of thumb is that the normal approximation is adequate when lambda exceeds about 20.
Recherches sur la probabilite des jugements en matiere criminelle et en matiere civile , Poisson, S.D. (1837) - Original derivation of the distribution
The Poisson model assumes three conditions: events occur one at a time (no simultaneous arrivals), the rate lambda is constant across the observation window, and events are independent of each other. Violating any of these changes the appropriate model.\n\nOverdispersion is the most common failure mode. If the observed variance in count data substantially exceeds the mean, the independence assumption is likely broken (e.g., goals in football cluster because teams pressing after a goal concede more). The negative binomial distribution is the standard alternative for overdispersed counts.\n\nUnderdispersion (variance less than mean) is rarer but occurs in data with regular spacing or inhibitory processes (e.g., a machine that cannot fire twice within a refractory window).
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This poisson distribution is an editorial illustration built to represent the concept accurately. Where it shows figures, they are typical or representative values chosen to make the relationship clear, not a single underlying dataset. The diagram and its explainer are reviewed and maintained centrally, and updated over time as understanding improves.
Rare-event counting in fixed intervals
The Poisson distribution models the count of events occurring in a fixed time or space interval when events happen independently at a constant average rate. A single parameter lambda controls both the mean and variance, which are equal.
Examples abound: the number of phone calls arriving at a switchboard in an hour, the count of manufacturing defects in a production run, or the number of customer arrivals at a service desk in a day. If lambda = 3, the expected count is 3, and the distribution spreads around that value.
Discrete shape and the rare-event approximation
Unlike continuous distributions, the Poisson is discrete, taking integer values 0, 1, 2, and so on. For rare events (small lambda), the distribution concentrates near zero with a rapid decay. For larger lambda, the distribution becomes more bell-shaped and approximates a normal distribution (another consequence of the central limit theorem).
Interestingly, the Poisson is the limiting distribution of binomial when n is large, p is small, and n times p equals lambda. This connection explains why rare events in large populations often fit a Poisson model better than a binomial model, despite both being valid.
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