Inspection Paradox

The Inspection Paradox (also known as length-biased sampling) arises in renewal processes when sampling is proportional to duration rather than discrete event counts. If gaps between events have variable lengths, an observer arriving at a uniformly random time is more likely to fall into a longer gap. Consequently, the mean length of the gap containing the arrival time is strictly greater than the ordinary average length of all gaps.

Experiment

This analytic module computes these stationary probabilities without running a finite-horizon random simulation. Try the following sequence to observe the paradox dynamically by interacting with the parameters on the main screen:

1. Set Long gap b to 9 and Long-gap fraction p to 0.1.
2. Observe that the Ordinary Mean E[L] is small (1.8), since 90% of gaps are length 1.
3. Notice that the Time Observed Mean E[L²]/E[L] is 5. Because the length 9 gaps are so massive, they take up 50% of the overall timeline (Time Observed Long Prob is 1/2), inflating the time-sampled mean far above the ordinary mean.
4. Change Long gap b down to 1. Notice the ordinary and time-observed metrics become perfectly equal (1 and 1). At b=1, the variance is zero and no length bias exists.

You can return to the module using this link: Inspection Paradox. It remains an excellent way to see the distinction between time-average and event-average metrics dynamically computed in the browser.