gamma distribution
The gamma distribution is a flexible, positive-valued, right-skewed family that is the actuary's go-to for moderately heavy claim sizes and waiting times. Picture not the wait until the first event but the wait until the third or fifth — the total of several exponential waits. That total is gamma-distributed, and by tuning its shape it can look like a sharp early peak, a gentle hump, or almost a bell.
It has two parameters, usually a shape α (alpha) and a scale θ (theta). The mean is αθ and the variance is αθ^2. When the shape α equals 1, the gamma collapses into a plain exponential; as α grows large, the gamma starts to resemble a normal curve. Summing α independent exponential waits, each with mean θ, gives a gamma — which is why it naturally models accumulated time or accumulated severity. It always lives on the positive side, which fits claim amounts that can never be negative.
In practice the gamma is a popular severity distribution for claim sizes that are skewed but not wildly heavy-tailed, and it is the response distribution of choice in many generalized linear models used for pricing. Its starring supporting role, though, is as the mixing distribution that turns a Poisson into a negative binomial: letting each policyholder's Poisson claim rate vary according to a gamma produces exactly the overdispersed counts seen in real portfolios. For the very heaviest tails, however — the largest catastrophe losses — the gamma is too light, and a Pareto or lognormal is preferred.
Model claim sizes with a gamma of shape α = 2 and scale θ = 1,500. The mean claim is αθ = $3,000 and the distribution is right-skewed: most claims are below the mean, with a tail of larger ones — a realistic shape for many lines.
The gamma flexibly fits skewed, positive claim sizes — a workhorse severity model.
The gamma has a relatively light (exponential-type) tail; for lines dominated by rare giant losses it understates extreme risk, so a heavier-tailed Pareto or lognormal is usually safer there.