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Concept

Poisson Distribution

The Poisson distribution models counts of events over a fixed exposure when events occur independently at a stable average rate.

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Poisson distribution

Formula

If X is Poisson with rate lambda, the probability of k events is given by the mass function below.

Probability mass function
P(X=k)=e−λλkk!,k=0,1,2,…P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!},\quad k=0,1,2,\ldots
Mean and variance
E[X]=λ,Var⁡(X)=λE[X]=\lambda,\qquad \operatorname{Var}(X)=\lambda

Worked Example

If a portfolio averages 2 claims per month, the probability of exactly 3 claims next month is about 0.180.

Claim count example
P(X=3)=e−2233!≈0.180P(X=3)=e^{-2}\frac{2^3}{3!}\approx 0.180

How This Appears On Exams

On Exam P, the Poisson distribution usually appears as an event-count model: claims per month, accidents per interval, calls per hour, or defects per unit exposure. On ASTAM and CAS statistics pages, it becomes part of frequency-severity thinking, compound models, and count modeling.

The key exam skill is not just remembering the probability mass function. You need to recognize when a count model is appropriate, know that the mean and variance are both lambda, and connect Poisson arrivals to exponential waiting times.

Statistics Connection

Poisson models become count regression when the expected count depends on predictors such as exposure, region, or policy characteristics.

References And Official Sources