Average wait for a bus every ten minutes

Question

Buses leave a stop exactly every $10$ minutes, like clockwork. If you turn up at the stop at a uniformly random moment, how many minutes do you expect to wait for the next bus?

Short answer

$5$ minutes. Arriving at a random moment, your wait is uniform on $[0,10]$ , whose mean is the midpoint $5$ .

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Lessons behind this question

Duration and Convexity: Measuring and Managing Interest Rate Sensitivity
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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