Probability and Mathematical Statistics
This domain tests the probability and statistical machinery behind every later pricing and reserving calculation: distribution theory, transforms, limit theorems, point estimation and hypothesis testing. Items are short and computational rather than proof-based, so speed with standard results matters more than derivations.
Distributions and transforms
- Know mgf/cgf techniques: sums of independent variables multiply mgfs; the mgf of a mixture is the weighted average of component mgfs.
- Use the CLT with continuity correction for aggregate claim counts, and the delta method for variances of transformed estimates, e.g. Var(g(X)) ≈ [g'(μ)]²Var(X).
- Recognise Poisson, gamma, lognormal, Pareto and compound distributions by their mean-variance structure.
Estimation and testing
- MLEs are invariant under reparameterisation; check second derivatives or boundary behaviour for maxima.
- Sufficiency via the factorisation theorem; Rao-Blackwell and Cramér-Rao give the benchmark variance.
- For tests, match degrees of freedom to estimated parameters, and use t rather than z when the variance is estimated from small samples.
Worked example
A screening test has prevalence 2%, sensitivity 95%, specificity 90%. Then P(+) = 0.02(0.95) + 0.98(0.10) = 0.019 + 0.098 = 0.117, so P(disease | positive) = 0.019/0.117 = 0.162, only about 16%. The low base rate dominates the accurate test.
Traps
- Confusing P(A|B) with P(B|A), and neglecting the base rate.
- Forgetting dependence when adding variances: Var(X+Y) = Var(X)+Var(Y)+2Cov(X,Y).
- Reporting an MLE without checking it lies inside the parameter space.
- Using normal critical values for small-sample variance estimates.
- Mixing up the mean and variance of a mixture: the mixture variance includes between-component variation.