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In: Statistics and Probability

Suppose that people arrive at a bus stop in accordance with a Poisson process with rate...

Suppose that people arrive at a bus stop in accordance with a Poisson process with rate λ. The bus departs at time t.

1. Suppose everyone arrives will wait until the bus comes, i.e., everyone arrives during [0, t] will get on the bus. What is the probability that the bus departs with n people aboard?

2. Let X be the total amount of waiting time of all those who get on the bus at time t. Find E[X]. (Hint: condition on the number of people on the bus.)

Suppose each person arrives at the bus stop will independently wait some time that has an exponential distribution with rate μ . If no bus arrives, he/she will leave the bus stop.

3. What is the probability that the bus departs with n people aboard? (Hint: Thinning a Poisson process, if a person arrives at the bus stop (arriving time uniformly distributed), calculate the probability that he/she will get on the bus)

4. If at time s (s < t), there are k people waiting at the bus stop. What is the expected number of customers who will get on the bus at time t ? (Note some people may leave the bus stop and some may arrive.)

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Expert Solution

orchestra answered 2 years ago
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