Question

1. (Sethna 1.3, see hints in textbook) On a highway, the average numbers of cars and buses going east are equal: each hour, on average, there are 12 buses and 12 cars passing by. The buses are scheduled: each bus appears exactly 5 minutes after the previous one. On the other hand, the cars appear at random: in a short interval dt, the probability that a car cornes by is dt/T , with T-5 minutes. An observer is counting the cars and buses. a) Verify that each hour the average number of cars passing the observer is 12 (b) What is the probability Pbus (n) that n buses pass the observer in a randomly chosen 10 minute interval? And what is the probability Par(n) that n cars pass the observer in the same tin e interval? (c) What are the probability distributions bus and car for the time interval Д between two successive buses and cars, respectively? What are the means of these distributions? (d) If another observer arrives at the road at a randomly chosen time, what is the probability distribution for the time Д she has to wait for the first bus to arrive? What is the probability distribution for the time she has to wait for the first car to pass by? What are the means of these distributions?

Answer 1

-25 tdl ky dt 乏-