Question

Question D

C. In Regular Bus City, there is a shuttle bus that goes between Stop A and Stop B, with no stops in between. The bus is perfectly punctual and arrives at Stop A at precise five minute intervals (6:00, 6:05, 6:10, 6:15, etc.) day and night, at which point it immediately picks up all passengers waiting. Citizens of Regular Bus City arrive at Stop A at Poisson random times, with an average of 5 passengers arriving every minute, and board the next bus that arrives (a) Suppose that you visit this city and that you arrive at Stop A at a time chosen uniformly at random from the times in a day. How long do you expect to have to wait until the next bus? (b) How many citizens of Regular Bus City do you expect to be on the bus that you take? In Poisson Bus City, there is a shuttle bus that goes between Stop A and Stop B, with no stops in between. The times at which the bus arrives at Stop A are a Poisson point process with one bus arriving every five minutes on average, day and night, at which point it immediately picks up all passengers waiting. Citizens of Poisson Bus City (like those of Regular Bus City) arrive at Stop A at Poisson random times, with an average of 5 passengers arriving every minute, and board the next bus that arrives. (c) Suppose that you visit this city and that you arrive at Stop A at a time chosen uniformly at random from the times in a day. How long do you expect to have to wait until the next bus? (d) How many citizens of Poisson Bus City do you expect to be on the bus that you take? (e) Are the following two statements true or false? If they are both true, explain in words the apparent discrepancy: (i) When you visit, buses in Poisson Bus City seem on average to come twice as slowly and to be twice as crowded as those in Regular Bus City (ii) In both cities, buses come on average every five minutes and people come on average five times per minutes, so that over the long haul there are 25 people per bus on average so buses are on average equally crowded in the two cities (c) If there are 10, 000 similar facts (each fact comes with same probabilities as above), how many of them do we expect that Jill knows but Alice doesn't (assuming that both are 10,000 days old)? Assume that for each given fact, the two Poisson random variables (number of times fact is heard by Alice and by Jill) are independent. (If the answer is small, then Jill should feel pretty lucky when one of these facts comes up while she is watching Jeopardy with Alice.)

Answer 1

Bus arrives at an average interval of 5 minutes

5 citizen arrives the bus station per minute

so 25 citizen arrives at bus station in every 5 minutes

citizen reached bus station immediately board .so all 25 citizens travel in that bus

so expected citizen in bus is 25