Probability and Distributions for Insurance

Every ratemaking, reserving, and reinsurance calculation an actuary performs rests on a small toolbox of probability distributions. This article surveys the frequency and severity families used to model claim counts and claim sizes, reviews moment-generating functions (MGFs) as a computational shortcut, and shows how mixing distributions produces the heavy-tailed, overdispersed behavior real loss data exhibits.

Frequency distributions

Claim counts are modeled with discrete, non-negative distributions. The workhorse is the Poisson: $P(N=n) = e^{-\lambda}\lambda^n/n!$, with $E[N]=\operatorname{Var}(N)=\lambda$ — the "equidispersion" property. Real portfolios are almost always overdispersed ($\operatorname{Var}(N) > E[N]$) because of parameter uncertainty across risks, so the negative binomial is the practical default: with parameters $r$ (or $\alpha$) and $\beta$, $E[N]=r\beta$ and $\operatorname{Var}(N)=r\beta(1+\beta) > E[N]$. The negative binomial arises naturally as a Poisson mixed over a gamma heterogeneity parameter — exactly the frequency analogue of Bühlmann credibility.

The binomial applies when there is a hard cap on the number of events (e.g., at most one claim per policy per peril in a simplified model): $P(N=n) = \binom{m}{n}q^n(1-q)^{m-n}$.

The $(a,b,0)$ class unifies Poisson, negative binomial, and binomial: each satisfies $P(N=n)/P(N=n-1) = a + b/n$ for constants $a,b$, which is the foundation of the Panjer recursion used to build aggregate loss distributions.

Severity distributions

Claim severity is continuous, positive, and right-skewed. Key families:

• Exponential: $f(x) = \theta^{-1}e^{-x/\theta}$, memoryless, $E[X]=\theta$, $\operatorname{Var}(X)=\theta^2$.
• Gamma: $f(x) = \dfrac{x^{\alpha-1}e^{-x/\theta}}{\Gamma(\alpha)\theta^\alpha}$, $E[X]=\alpha\theta$, $\operatorname{Var}(X)=\alpha\theta^2$; flexible shape, thinner tail than lognormal for large $\alpha$.
• Lognormal: $\ln X \sim N(\mu,\sigma^2)$, so $E[X]=e^{\mu+\sigma^2/2}$ and $\operatorname{Var}(X) = e^{2\mu+\sigma^2}(e^{\sigma^2}-1)$. Popular for liability severity because it is easy to fit via logs and has a moderately heavy tail.
• Pareto (single-parameter and two-parameter): two-parameter with scale $\theta$ and shape $\alpha$ has survival function $S(x) = \left(\dfrac{\theta}{x+\theta}\right)^\alpha$; mean $E[X]=\theta/(\alpha-1)$ for $\alpha>1$ and infinite variance for $\alpha \le 2$ — the standard model for excess/umbrella layers because of its regularly varying tail.
• Weibull: $S(x) = e^{-(x/\theta)^\tau}$; $\tau<1$ produces heavier-than-exponential tails.

Moments and the moment generating function

The MGF $M_X(t) = E[e^{tX}]$ encodes all moments via $M_X^{(k)}(0) = E[X^k]$, and — crucially for aggregate loss modeling — the MGF of a sum of independent variables is the product of the individual MGFs, while the MGF of a compound sum $S = \sum_{i=1}^N X_i$ satisfies

$M_S(t) = M_N(\ln M_X(t)).$

This single identity is why frequency and severity are modeled separately and then combined: it lets us compute $E[S]$ and $\operatorname{Var}(S)$ from the frequency and severity moments alone, without ever deriving the distribution of $S$ itself. When $N$ and $X$ are independent,

$E[S] = E[N]E[X], \qquad \operatorname{Var}(S) = E[N]\operatorname{Var}(X) + \operatorname{Var}(N)E[X]^2.$

Mixture distributions

A mixture blends component distributions, either discretely ($f(x) = \sum_k w_k f_k(x)$) or continuously (a distribution's parameter is itself random, e.g. the Poisson-gamma mixture giving negative binomial). Mixtures are the natural language for:

• Heterogeneous portfolios: each policyholder has their own $\lambda$, drawn from a population distribution — this is exactly the credibility problem.
• Fat tails from "normal" components: a lognormal-Pareto spliced model uses lognormal for the body of the severity distribution and Pareto above a threshold, matching empirical claims data far better than either alone.
• Zero-inflation: a point mass at zero mixed with a Poisson or negative binomial models portfolios with many risks that never report a claim.

Worked example

A book of 10,000 policies has claim frequency $N \sim \text{Poisson}(\lambda = 0.08)$ per policy and, given a claim occurs, severity $X \sim \text{Lognormal}(\mu = 8.5, \sigma = 1.2)$ (in dollars). Find the expected aggregate loss per policy and its standard deviation.

Frequency moments: $E[N] = 0.08$, $\operatorname{Var}(N) = 0.08$.

Severity moments: $E[X] = e^{8.5 + 1.2^2/2} = e^{8.5+0.72} = e^{9.22} \approx \$10{,}085.$ $\operatorname{Var}(X) = e^{2(8.5)+1.44}\left(e^{1.44}-1\right) = e^{18.44}(3.2201) \approx 1.0025\times10^{8} \times 3.2201 \approx 3.228\times10^{8}.$ So $E[X^2] = \operatorname{Var}(X) + E[X]^2 \approx 3.228\times10^8 + 1.017\times10^8 = 4.245\times10^8$, and the severity standard deviation is \sqrt{3.228\times10^8}\approx \17{,}967$.

Aggregate per policy: $E[S] = 0.08 \times 10{,}085 \approx \$806.8.$ $\operatorname{Var}(S) = 0.08 \times 3.228\times10^{8} + 0.08 \times 10{,}085^2 \approx 2.582\times10^7 + 8.137\times10^6 \approx 3.396\times10^7.$ $\text{SD}(S) \approx \$5{,}828.$

Notice the standard deviation ($5,828) dwarfs the mean ($807) — a coefficient of variation over 7 — which is exactly why individual-policy pricing needs credibility weighting against class-wide experience rather than raw policy-level loss ratios.

Pitfalls

• Confusing per-claim and per-policy moments. Frequency is "claims per exposure," severity is "dollars per claim"; mixing them up double- or under-counts variance.
• Assuming independence of frequency and severity when large claims correlate with catastrophe-driven frequency spikes (e.g., a hurricane year has both more claims and larger claims).
• Fitting lognormal by ordinary least squares on raw dollars instead of MLE on logs — this understates the tail.
• Ignoring truncation and censoring: policy limits and deductibles truncate/censor the observed severity distribution; fitting to raw claims data without adjusting for the retention biases parameter estimates.
• Using Pareto with $\alpha \le 1$ and still quoting a "mean" — the mean does not exist.

Exam relevance

Frequency/severity families, MGFs, and mixtures are core to CAS Exam MAS-I and SOA Exam STAM/SRM, and re-appear throughout CAS Exams 5, 8, and 9 whenever a loss distribution must be selected or an aggregate distribution built.