Poisson Distribution

Study guides: Exam P-1, Exam MAS-I

The Poisson Distribution X \sim \text{Poi}(\lambda) models the number of events occurring in a fixed interval of time or space when events happen independently at a constant average rate \lambda.

P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}

k = 0, 1, 2, \ldots

\text{where } \lambda > 0 = \text{the rate (mean number of events)}

• A defining property is E[X] = \text{Var}(X) = \lambda — the mean and variance are equal
• It arises as the limit of \text{Bin}(n, p) as n \to \infty and p \to 0 with np = \lambda fixed
• It is the standard model for claim counts in actuarial science

Probability of Zero Claims in a Month Claims arrive at an average rate of \lambda = 3 per month. What is the probability of receiving no claims in a given month?

Answer P(X = 0) = \frac{e^{-3} \cdot 3^0}{0!} = e^{-3} \approx 0.0498 There is approximately a 5% chance of a claim-free month.