Overview

This topic opens Stage 2 of Probability II: the move from discrete to continuous time. The organising idea is the exponential distribution's memoryless property, which plays exactly the role for continuous-time processes that the Markov property played for discrete chains — it is the single assumption that makes the Poisson process tractable, and its removal (moving to general i.i.d. inter-arrival times) is exactly what defines the more general renewal process this topic closes with. The topic follows Durrett's Chapters 2–3 closely: build the exponential distribution's key properties, define the Poisson process three equivalent ways, develop the operations (compounding, thinning, superposition) that build richer models from it, then generalise to renewal processes and derive the two limit theorems (elementary and key renewal theorems) that make renewal theory usable in practice, closing with the inspection paradox and the renewal-reward theorem.