Distributions Atlas: Actuarial View
Actuarial work uses a small fixed set of distributions, used in two roles: frequency models for counts and severity models for amounts. The atlas organizes them by role and by tail weight so the right model is easy to pick before any algebra starts.
- Role
- Concept
- Level
- Core
- Time
- Reference
- Freshness
- Stable
Two Roles, Not One List
Every distribution that appears on Exam P, FAM, or ASTAM is doing one of two jobs. A frequency model counts events over an exposure: claim counts per policy, accidents per quarter, defaults per cohort. A severity model assigns an amount to each event: claim size, time to failure, loss amount.
The aggregate loss S = X_1 + X_2 + ... + X_N then has N drawn from a frequency model and each X_i drawn from a severity model. Klugman organizes Loss Models around this split, and the SOA short-term syllabus is built the same way. Memorizing distributions without their role is the most common reason candidates pick the wrong model under exam pressure.
Frequency Models
Frequency models are discrete and supported on the non-negative integers. The (a, b, 0) class on the Loss Models tables is exactly Poisson, Binomial, and Negative Binomial (which contains the Geometric as r = 1). The (a, b, 1) extension adds zero-modified and zero-truncated variants.
Poisson is the baseline for independent rare events. Binomial is the right model when the number of trials is fixed. Negative Binomial is the right model when Poisson under-states variance — that overdispersion is exactly the gamma-mixed-Poisson story.
- Poisson: mean = variance. Pick it for rare events at a stable rate.
- Binomial: variance < mean. Pick it when m trials and probability q per trial are given.
- Negative Binomial: variance > mean. Pick it when counts are overdispersed.
- Geometric: NegBin with r = 1. A waiting-time-in-trials story.
- Hypergeometric: sampling without replacement from a finite population. Outside the (a, b, 0) class.
Severity Models
Severity models are continuous and positive. The actuarial short list is Exponential, Gamma, Lognormal, Pareto, Weibull, and Inverse Gaussian. Each has its own shape behavior and its own tail.
Two parameterization conventions matter on the SOA tables. Gamma, Exponential, and Pareto use a scale parameter (θ). Weibull also uses a scale parameter (θ) and a shape parameter (τ). Lognormal uses the mean and standard deviation on the log scale (μ and σ). New pages call out the parameter convention explicitly, because mixing rate-form and scale-form is the single biggest source of moment-error on ASTAM.
Tail Weight: Light, Medium, Heavy
Tail weight matters because reserves, reinsurance, and capital decisions are dominated by what the upper tail does. A common ranking from lightest to heaviest, holding the mean fixed:
- Light tail: Normal (truncated for severity work), Exponential, Gamma with shape > 1, Weibull with τ > 1, Inverse Gaussian.
- Medium tail: Gamma with shape ≤ 1, Weibull with τ < 1, Lognormal with small σ.
- Heavy tail: Lognormal with large σ, Pareto with small α, Weibull with τ much less than 1.
Exam Coverage Matrix
Exam P names binomial, geometric, hypergeometric, negative binomial, Poisson, and uniform on the discrete side, plus beta, exponential, gamma, normal, and uniform on the continuous side. Exam P does not test Pareto, Lognormal, Weibull, or Inverse Gaussian.
FAM adds severity and aggregate modeling: Exponential, Gamma, Pareto, Lognormal, Weibull, and Inverse Gaussian appear as severity options, and Poisson, Binomial, and Negative Binomial appear inside the (a, b, 0) frequency class. Compound Poisson aggregates begin here.
ASTAM extends the same model list into estimation, intervals, goodness-of-fit, model selection, and credibility. ASTAM is where every distribution on the SOA Loss Models table is fair game.
How To Pick A Model Quickly
Start with the role. Is the question about a count or an amount? That alone eliminates half the list.
Then ask about the shape of the story. Fixed number of trials means Binomial. Random number of events at a stable rate means Poisson. Overdispersed counts mean Negative Binomial. Continuous waiting time means Exponential or Gamma. Continuous amount with a known heavy tail signal means Lognormal or Pareto. Continuous lifetime with hazard-rate intuition means Weibull.
Only at that point should you pull a formula. Picking the wrong model and then computing carefully is still wrong; picking the right model and then computing roughly often earns most of the points.