Next Bus Fuel Stop Wait Time
Expected waiting time at bus stop is a medium quant interview question on Conditional Expectation, reported to have been seen at Jane Street.
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This quant interview question is about conditional expectation under a nontrivial arrival process. The twist is that the effective intervals between buses are random and heavy-tailed, so your arrival time is more likely to fall in long gaps than short ones. It lives at the intersection of stochastic processes, discrete probabilities, and intuition about time-averaged versus event-averaged views of a system.
It trains conditional probability, law of total expectation, and a precise understanding of the inspection paradox. You must identify the relevant probability space, weight scenarios correctly by how much time they occupy, and compute an expected waiting time in that biased sampling regime. It also demands clean algebra and comfort translating word problems into rigorous random variables.
This matters for quant interviews because markets, order flow, and latency all exhibit similar selection biases. Quant prep that ignores these paradoxes leads to bad intuition about fill times, queueing, and risk. Top trading firms use questions like this to see whether you notice hidden conditioning, avoid naïve symmetry arguments, and reason correctly about random times, not just random events.
What it tests
When arrival times are uniformly random and service intervals are of variable length, the probability of arriving during a given interval is proportional to the interval's length. This is known as the inspection paradox or waiting time paradox: longer intervals are more likely to be observed simply because they occupy more of the timeline. The expected waiting time is then the average of half the interval lengths, weighted by the probability of being in each interval (which itself is proportional to the interval's length). This principle holds because, over a long period, the fraction of time spent in each type of interval matches its duration relative to the total cycle, not just its frequency. Thus, rare but long intervals can dominate the expected waiting time even if they occur infrequently.
Practise this question with written feedback, or hear it in a spoken mock interview.
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