Actuarial Statistics (CS1)
CS1 Actuarial Statistics tests whether you can turn an insurance problem into a probability model, fit it, and interpret the output. Questions mix derivation, computation and explanation, so show why a formula applies, not just the formula.
Distribution toolkit
Know when to use the pgf G(z)=E[z^X], the mgf or the cgf, and how each behaves for sums of independent variables. For aggregate claims S=ΣX_i with N independent of the X_i: E[S]=E[N]E[X] and Var(S)=E[N]Var(X)+Var(N)E[X]^2; use Panjer recursion when N is Poisson, binomial or negative binomial and severities are lattice-valued. For transformations include the Jacobian; the maximum of n iid variables has density n f(x)[1-F(x)]^{n-1}.
Inference and regression
Maximum likelihood drives most of the paper: write the likelihood, take logs, differentiate and check the second derivative. Use invariance for functions of parameters and asymptotic normality via observed or expected information. In a GLM, g(μ)=x'β and Var(Y)=φV(μ)/w; the deviance compares nested models, and when φ>1 compare deviance differences with φχ², not χ².
Survival and stochastic processes
Right-censored observations contribute f(t) at death and S(t) at censoring. Kaplan–Meier multiplies (1−d/n) at each death time using the risk set n. Cox regression leaves the baseline hazard unspecified and estimates β by partial likelihood under proportional hazards. For Markov chains use P^n and the stationary distribution π=πP; for a Poisson process use exponential interarrival times, thinning and superposition.
Worked example
Example: n independent exponential lifetimes, d deaths at t_i and c censored times. The likelihood is Π_deaths λe^{−λt_i} × Π_censored e^{−λt_i} = λ^d e^{−λT}, with T the total time at risk. Setting d log L/dλ = d/λ − T = 0 gives λ̂ = d/T; the second derivative −d/λ² < 0 confirms a maximum. With 3 deaths and total exposure 12, λ̂ = 0.25 per year.
Traps
Traps: confusing pgf with mgf; omitting the Jacobian; using the full sample size instead of the risk set in Kaplan–Meier; reporting a Cox coefficient as a hazard rather than a log hazard ratio; assuming the stationary distribution equals the limiting distribution for a periodic chain; and forgetting that a Poisson process rate is not a probability.