When did the second of four calls arrive?

Question

Calls arrive at a hotline as a Poisson process during an $8$ -hour shift, $[0, 8]$ hours. You are told that exactly $4$ calls came in during the shift.

Given this, what is the expected arrival time (in hours) of the $2$ nd call?

Accepted Answer

There is a striking fact about Poisson processes: once you condition on the number of arrivals in an interval, the arrival times behave exactly like that many points dropped independently and uniformly at random on the interval, then sorted. The rate \lambda disappears entirely once the count is fixed. So given 4 calls in [0, 8], the four call times act like 4 uniform points on [0, 8], sorted from earliest to latest. The 2nd call is the 2nd smallest of those. For n uniform points on an interval of length T, the sorted values split the interval into n + 1 gaps that are equal on average. The k-…

When did the second of four calls arrive?

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