A discrete probability distribution on the nonnegative integers that models the count of independent rare events occurring in a fixed interval, determined by a single rate parameter.

Principle

Counts in disjoint subintervals are independent and the probability of more than one event in an infinitesimal subinterval is negligible; event rate scales linearly with interval length.

Demonstration

If events occur with constant average rate λ per unit interval, the probability of observing k events in one interval is e^{-λ} λ^{k} / k!.

Misapplication

Using the distribution for data with strong overdispersion or dependency between events without adjusting the model leads to biased inference.

Consequence

Provides a simple one-parameter model for counts that facilitates likelihood-based estimation and Poisson-based inferential tools for rates.

Reversal

Instead of modeling counts from a rate, fix counts and infer variability by treating the rate parameter as unknown and estimating it from data.

Boundary

Appropriate for independent, low-probability events with constant rate; not suitable when events cluster, have memory, or when counts are bounded with structural zeros.

Semantic Tension

Competes with the negative binomial for overdispersed counts and with binomial when the number of trials is known and probabilities are not small.

Synthesis

A canonical discrete law that links a constant average occurrence rate to the distribution of event counts in an interval under independence and rarity assumptions.