Waiting for a truck on a lonely highway
On a highway, trucks pass a given spot at random moments, independently of each other, at an average rate of one every 10 minutes (a Poisson process). The gaps between successive trucks are therefore exponentially distributed with mean 10 minutes.
Your car breaks down at a random moment.
- How long do you expect to wait for the next truck?
- How long ago, on average, did the previous truck pass?
- Given your answers, the gap between the previous truck and the next one averages 20 minutes. But gaps average 10 minutes. Explain the contradiction.
Show a hint
For part 1, use memorylessness: the process does not know how long it has been since the last truck. For part 3, ask whether the gap you happen to fall into is a typical gap.
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