Poisson frequency model
/ PWAH-sson /
Think of accidents arriving like raindrops on a windowpane: rare, independent, scattered randomly through time, with no two falling at exactly the same instant. If claims behave like that — each one a small chance, lots of opportunities, no clustering — then the number arriving in a year follows a Poisson distribution. It is the simplest and most natural starting model for counting claims.
The Poisson distribution has a single parameter, lambda, which is both its mean and its variance. The probability of exactly k claims is P(N=k) = e^(-lambda) times lambda^k divided by k-factorial. So if lambda = 2 (two claims expected per year), P(N=0) = e^(-2) is about 0.135, P(N=1) is about 0.271, and so on. A beautiful property: if you combine independent Poisson risks, the total is again Poisson with lambda equal to the sum — so a portfolio of many small Poisson policies is itself Poisson, which makes it wonderfully tractable.
The Poisson is the default frequency model in non-life insurance and the foundation of the compound Poisson aggregate model. Its great convenience is also its limitation: it forces variance to equal the mean. Real claim counts are often over-dispersed — the variance exceeds the mean, because policyholders differ in risk and some are accident-prone. When you see that, the Poisson under-states variability and a negative binomial (a Poisson with a random, gamma-distributed lambda) is the standard fix. The Poisson also assumes a constant rate and independence, which a sudden hailstorm causing many simultaneous claims clearly violates.
A small workshop expects on average 3 minor claims a year, so an actuary models its claim count as Poisson with lambda = 3. The model then says the chance of a claim-free year is e^(-3), about 5%, while the chance of 5 or more claims is around 18%. Because the mean and variance are both 3, the standard deviation of the count is the square root of 3, about 1.73.
One parameter lambda sets both the average and the spread of the claim count.
The Poisson's defining constraint is mean = variance. If observed claim counts vary more than that (over-dispersion, very common in practice), the Poisson understates risk and you should switch to a negative binomial rather than force the data to fit.