Why does the typical wait feel longer?
Sampling a process at a random time weights intervals by their length.
Inside this model
Short intervals last 2 minutes and long intervals last 10. The mean interval is the frequency-weighted average; a random arrival sees the length-biased average E[L²]/E[L]. Its expected residual wait is E[L²]/(2E[L]) for uniformly sampled time inside intervals.
Out in the world
A practical use
A passenger arriving without a timetable experiences a different average bus interval than the average printed gap.
A useful lens. Not a universal law.
- This is an ideal stationary renewal process with two interval lengths; scheduled arrivals change the result.
Further reading
Explore the original research or the teaching reference behind this experiment.
W. Feller — An Introduction to Probability Theory, Vol. II ↗