Probability
Random Incidence
Randomly arriving observers are biased toward longer intervals
Arrival process
True vs observed interval distribution (500 random pins)
The paradox
- Longer intervals take up more time, so a random observer is more likely to land in one.
- The observed average is always larger than the true average interval length.
- For Poisson processes, $\mathbb{E}[\text{observed}] = \frac{2}{\lambda}$ while $\mathbb{E}[\text{true}] = \frac{1}{\lambda}$. Exactly double.
- This applies everywhere: bus wait times, class sizes, packet interarrival. The "inspection paradox."