等卡车
Waiting for a Truck
题目详情
某公路上卡车到达服从泊松过程，强度 ，即相邻卡车到达间隔服从均值 10 分钟的指数分布。
你的车在随机时刻抛锚，你在路边等待下一辆卡车。
- 你期望还要等多久？
- 平均来说，上一辆卡车在多少分钟前经过？
提示：指数分布无记忆性；但不要简单答 5 分钟。
On a given highway, trucks arrive at the station according to a Poisson process with Lambda = 0.1/minute. This means that after a truck is just passed, the time for the next truck to arrive is an exponential random number with average arrival time of 10 minutes. Your car just broke on this highway, and you are waiting for the next truck for hitchhiking, what is your expected waiting time? On average how many minutes ago the last truck left?
Hint
It's not 5 minutes. Exponential distribution has memoryless property.
解析
两个答案都为 10 分钟。
指数分布具有无记忆性：从任意时刻起，距离下一次到达的剩余时间分布仍是均值 的指数分布。
同样地，“向后回溯到上一次到达”的期望也为 10。
这与“平均间隔 10”不矛盾：你在随机时刻更容易落在较长的间隔里（inspection paradox）。
Original Explanation
Solution
Using memoryless property of exponential distribution, the expected waiting time is 10 minutes. This also holds backwards, hence the expected time last truck passed is also 10 minutes. But this does not violate the total inter-arrival time of 10 minutes, because if your car breaks at a random time, you are more likely to be in long interval than a short one. I don't have time today, remind me to post a more mathematical explanation.