How often do events happen? The Poisson frequency model
Interactive Poisson distribution explorer for insurance claim counts and hospital admissions. Move the rate λ and watch P(X=k), the mode, and the probability of zero or extreme counts update live.
Frequency models answer the first question in any risk calculation: how many events should we expect in a fixed window? Insurance claims per month, hospital admissions per day, customer arrivals per hour: when events are independent and occur at a stable average rate λ, the Poisson distribution gives the probability of observing exactly k of them.
The Poisson has one parameter, and its variance equals its mean. That single fact explains why both quiet stretches and clusters are normal. At the default of λ = 5 claims a month, a month with no claims happens about 0.7% of the time, and a month with more than ten claims (over twice the average) about 1.4% of the time. Neither is, on its own, evidence that the rate has changed.
In actuarial pricing the Poisson is the baseline for claim frequency. When observed counts are more dispersed than it allows (variance greater than the mean), practitioners move to the Negative Binomial or another mixed Poisson model. The module offers two settings, monthly insurance claims and daily hospital admissions, so you can build intuition for the baseline before adding that complexity.
Poisson probability mass function
P(X = k) = e^{−λ} · λ^k / k!
λ is the expected number of events in the window and k = 0, 1, 2, … is the count observed; k! is k factorial (1 × 2 × … × k).
Frequently asked questions
When is the Poisson distribution appropriate for claim counts?
When events are independent, occur one at a time, and the underlying rate is stable over the window. If claims cluster (storms, epidemics) or the portfolio mixes very different risks, counts are over-dispersed and a Negative Binomial or mixed Poisson fits better.
What is the probability of zero claims under a Poisson model?
P(X = 0) = e^{−λ}. With λ = 2 expected claims it is about 13.5%; with λ = 5 it falls to about 0.7%.
Why does the Poisson variance equal its mean?
It follows from the distribution's derivation as the limit of a Binomial with many trials and a small success probability. In practice it is the diagnostic to check: if the sample variance of counts is much larger than the sample mean, the Poisson understates how often extreme counts occur.
Interactive explorer
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