Simulate and analyze point processes: Poisson, renewal (general inter-arrival distribution), Hawkes (self-exciting). Compare inter-event distributions, counting processes N(t), and the inspection paradox.
Poisson process: memoryless inter-arrivals T~Exp(λ). Renewal theorem: N(t)/t → 1/μ as t→∞. Inspection paradox: expected inter-arrival containing t is 2/λ (twice the mean!) — longer intervals more likely to contain t. Hawkes process: λ(t) = μ + Σ_{t_i