Exam STAM Formula Sheet: Key Distributions and Formulas
A comprehensive reference of essential distributions, formulas, and relationships for Exam STAM preparation.
Key Severity Distributions
Exponential: f(x) = (1/theta)*exp(-x/theta), E[X] = theta, Var(X) = theta^2. Pareto: f(x) = alpha*theta^alpha / (x+theta)^(alpha+1), E[X] = theta/(alpha-1) for alpha > 1. Lognormal: if Y = ln(X) ~ N(mu, sigma^2), then E[X] = exp(mu + sigma^2/2). Gamma: E[X] = alpha*theta, Var(X) = alpha*theta^2. Weibull: S(x) = exp(-(x/theta)^tau). These are the most commonly tested severity distributions. Know their CDFs, survival functions, limited expected values, and moment generating functions.
Key Frequency and Aggregate Formulas
Poisson: E[N] = Var(N) = lambda. Negative Binomial: E[N] = r*beta, Var(N) = r*beta*(1+beta). Compound model: E[S] = E[N]*E[X], Var(S) = E[N]*Var(X) + Var(N)*(E[X])^2. Normal approximation: P(S <= s) is approximately Phi((s - E[S])/sqrt(Var(S))). Credibility: Z = n/(n+k), k = v/a. Chain ladder CDF = product of age-to-age factors. Keep these formulas accessible throughout your STAM preparation for quick reference during practice problems.