##  [Poisson Distribution](/poisson-distribution-0) 

 Definition

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

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

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

Misapplication

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

 

 

 

 

 





## Consequence

Consequence

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

 

 

 

 

## Reversal

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

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

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

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.