Poisson Distribution: Derivation, Properties, and Insurance Applications

Definition and Derivation

The Poisson distribution models the count of events occurring in a fixed interval when events happen independently at a constant average rate. If X ~ Poisson(lambda), the PMF is P(X = k) = e^(-lambda) * lambda^k / k! for k = 0, 1, 2, ... Both the mean and variance equal lambda.

The Poisson arises as a limit of the Binomial(n, p) when n is large, p is small, and lambda = np is moderate. This approximation is useful when computing exact binomial probabilities is impractical. For Exam P, if n >= 50 and p <= 0.05, the Poisson approximation is generally appropriate.

Key Properties

The MGF of X ~ Poisson(lambda) is M(t) = exp(lambda(e^t - 1)). A critical property: the sum of independent Poisson random variables is Poisson. If X ~ Poisson(lambda1) and Y ~ Poisson(lambda2) independently, then X + Y ~ Poisson(lambda1 + lambda2). This makes the Poisson distribution particularly convenient for combining claim counts from independent lines of business.

The Poisson distribution is right-skewed, with skewness = 1/sqrt(lambda). As lambda increases, the distribution becomes approximately normal.

Insurance Applications

In insurance, the Poisson distribution is the standard model for claim frequency. If an insurer expects 3 claims per month on average, the probability of 0 claims is e^(-3) = 0.0498, and the probability of more than 5 claims is 1 - P(X <= 5). The Poisson assumption underlies many actuarial models including aggregate loss distributions and credibility theory. Exam P problems often combine Poisson frequency with a severity distribution to compute expected total losses.