Probability and Statistics
Distributions and moments
Master the frequency family (binomial, Poisson, negative binomial) and the severity family (exponential, gamma, lognormal, Pareto). Be able to produce means, variances, and moment generating functions on demand, and to recognize a distribution from its mgf or variance-to-mean ratio. The variance of a sum requires independence (or explicitly modeled covariance) - known marginals alone are insufficient.
Conditioning and Bayes
Use the law of total probability and the conditional variance formula freely. Bayesian updating follows posterior ∝ prior × likelihood; the Poisson–gamma and binomial–beta pairs are conjugate, so memorize the updated parameters rather than re-deriving them under time pressure.
Estimation and inference
Know maximum likelihood (log-likelihood, derivative, boundary cases), method of moments, confidence intervals, and the size/power distinction in hypothesis tests. Watch parameterizations: gamma with rate versus scale, Poisson mean versus rate times exposure, and truncated or censored likelihoods where the denominator changes.
Worked example
A portfolio is 70% good risks (claim probability 0.05) and 30% bad risks (0.20). One claim is observed. Then
P(bad | claim) = 0.30(0.20) / [0.30(0.20) + 0.70(0.05)] = 0.06/0.095 = 0.632.
Traps
Confusing standard error with standard deviation, forgetting the continuity correction on discrete approximations, and treating a censored observation as if it were a full observation are the most common errors.