The gap you land in 2

What this Poisson-process interview question tests

This is a medium-difficulty statistics question that explores how renewal processes behave when you arrive at a random time. It's the kind of problem quant trading desks use to assess whether a candidate understands the counterintuitive properties of Poisson arrivals and can apply them under time pressure.

The core challenge is recognizing that arrival at a uniformly random instant places you in a gap whose expected length depends on the rate parameter of the process—but not in the way naive reasoning might suggest. Working through this requires understanding the inspection paradox (or waiting-time paradox) and how to condition on the arrival process. Candidates who set up the problem cleanly, identify the relevant probability distribution, and apply linearity of expectation typically score well.

• Poisson process properties and memorylessness
• The inspection paradox and renewal theory
• Conditioning on rate and arrival time