Gamma Distribution

A gamma random variable X with shape α > 0 and scale θ > 0 takes positive values with the PDF below. The SOA Loss Models tables use the scale parameterization (α, θ). Some statistics texts use the rate parameterization (α, λ) with λ = 1/θ; the two are equivalent.

Confusing scale and rate is the single biggest source of moment-error on FAM and ASTAM. The exam table is the source of truth: under scale θ, the mean is αθ. Under rate λ, the mean is α/λ. Pick one and stay in it.

When α = 1 the gamma collapses to an exponential with mean θ. When α is an integer n, the gamma is the distribution of the sum of n independent exponentials with the same scale, and it is the waiting time to the n-th event in a Poisson process. This is the Erlang distribution.

When α < 1 the density is unbounded at zero and the distribution is heavily right-skewed. When α > 1 the density has a mode at (α − 1)θ and the distribution becomes more symmetric as α grows. The CV is 1/√α, so increasing the shape parameter directly tightens the relative spread.

Three distinct roles. First, it is a flexible severity model with light to medium tail, often used for moderate-sized claim amounts. Second, it is the mixing distribution that produces the negative binomial when used as a prior on a Poisson rate; see /concepts/geometric-negative-binomial/ for the identity. Third, it is the conjugate prior for the Poisson rate parameter in Bayesian credibility, which makes posterior computation closed-form.

On Tweedie models, the compound Poisson aggregate with gamma severities is a Tweedie distribution with power parameter strictly between 1 and 2. This is the standard pure-premium GLM in personal-lines insurance.

With shape α known, the gamma scale MLE is the sample mean divided by α. This is the cleanest closed form and is worth memorizing for ASTAM.

With both shape and scale unknown, the log-likelihood equations involve the digamma function and have no closed-form solution. Numerical optimization (Newton-Raphson or scoring) is used in practice. ASTAM problems usually fix α and ask for θ̂.

Claim severities are modeled as gamma with shape α = 2 and unknown scale θ. Five observed claims are 500, 800, 1,200, 600, and 1,100. The sample mean is 840. The MLE for the scale is θ̂ = 840 / 2 = 420.

The fitted mean is α θ̂ = 840 (it must match the sample mean by construction) and the fitted variance is α θ̂^2 = 2 × 420^2 = 352,800. The fitted CV is 1/√2 ≈ 0.707, which is a feature of the model choice rather than the data.

If X_1 is Gamma(α_1, θ) and X_2 is Gamma(α_2, θ) and they are independent with the same scale, then X_1 + X_2 is Gamma(α_1 + α_2, θ). Convolution is closed only when the scale is shared.

Three independent claim-payment delays with α = 1 and θ = 30 days sum to a Gamma(3, 30) waiting time. Mean is 90 days; variance is 2,700 days^2. This is also the Erlang(3, scale = 30) distribution and is the distribution of the third event in a Poisson process with rate 1/30 per day.

Suppose annual claim counts for one policyholder are Poisson with rate Λ, and Λ has a gamma prior with shape α_0 and scale θ_0. After observing total claims of T in n years, the posterior of Λ is gamma with shape α_0 + T and scale θ_0 / (1 + nθ_0).

The posterior mean is a credibility-weighted blend of the prior mean α_0θ_0 and the observed rate T/n. The credibility weight is exactly Z = nθ_0 / (1 + nθ_0), which matches the Bühlmann credibility formula for this setup. Klugman Loss Models develops this in the conjugate-prior section.