The gap you land in 1

What this waiting-time statistics question tests

This is a classic problem in probability and statistics that appears in quantitative finance interviews. It tests whether you understand the memoryless property of the Poisson process and how arrival times behave when you sample at a uniformly random moment rather than relative to a known event.

The scenario is deliberately simple: you join an ongoing stream of events (trades) that arrive at a constant average rate, but you have no knowledge of when the last event occurred. The key insight is recognizing that the waiting time until the next event is not simply the average inter-arrival interval—the distribution of your wait depends critically on where you land in the cycle. This type of reasoning is essential for market microstructure problems, order-book dynamics, and latency analysis.

• The memoryless property and why it matters for Poisson processes
• Intuition behind the inspection paradox
• Converting between rates (events per hour) and time scales