Derivation — The Inspection Paradox: Excess Life, Age, and Size-Biased Sampling

If bus inter-arrival times average 10 minutes, how long should a passenger arriving "at a random time" expect to wait for the next bus? The naive answer — 5 minutes, half the average gap — is wrong in general, and the true answer can be arbitrarily larger. This is the inspection paradox, and its resolution is a genuinely important, non-obvious fact about renewal processes: an interval sampled by "landing inside it at a random time" is systematically longer than a typical interval, because long intervals are more likely to contain a random inspection point than short ones. This note derives the exact limiting distribution of the age and residual life at a random inspection time, using a renewal-reward argument built directly on the renewal equation machinery.

Learning objectives. By the end of this note you should be able to: