Why does the bus always seem late?
You're more likely to arrive during a long gap than a short one!
Every 10 minutes exactly
Average 10 min (Poisson)
Interval: 10 min
Expected wait: 5 min
Average interval: 10 min
Expected wait: 10 min!
Same average frequency, but DOUBLE the wait time with irregular arrivals!
When you arrive at a random time, you're more likely to land in a long gap than a short one—simply because long gaps take up more time!
Imagine buses sometimes come 5 minutes apart and sometimes 15 minutes apart (averaging 10 min). The 15-minute gaps are three times longer, so you're three times more likely to arrive during one of them!
For regular buses with interval T, your expected wait is simply:
For irregular buses (Poisson process) with mean interval T:
The irregular case has double the expected wait! This is because the variance in arrival times creates longer gaps that "trap" random arrivals.
The inspection paradox appears everywhere in length-biased sampling:
This paradox explains why public transit often feels worse than statistics suggest. If buses are supposed to come every 10 minutes on average, passengers experience much longer waits during peak variance times.
Solution: Regularity matters more than frequency! A bus every 12 minutes like clockwork beats one that averages 10 minutes but varies from 5 to 20.
Whenever you sample something proportional to its size/duration, you oversample the large ones. This is called the inspection paradox or size-biased sampling.
The higher the variance, the worse your experience compared to the average!