Given three buses came, when did the first arrive?

Question

Buses arrive as a Poisson process during a one-hour window $[0, 60]$ minutes. You are told that exactly three buses arrived during the hour.

Given this, what is the expected arrival time of the first bus?

Accepted Answer

There is a beautiful fact about Poisson processes: conditioned on the number of arrivals n in an interval, the arrival times are distributed exactly like n independent uniform points on that interval, then sorted. The rate \lambda doesn't even appear once you've conditioned on the count. So given 3 arrivals in [0, 60], the three arrival times behave like the order statistics of 3 uniform draws on [0, 60]. The first arrival is the minimum of three uniforms. Using the equal-gaps result for uniforms (3 points split the interval into 4 equal-in-expectation gaps), the minimum sits at E[	ext{first…

Given three buses came, when did the first arrive?

Your answer

More Probability questions