wv "nw" "UA N 'AW 7. Buses arrive at a station according to a Poisson process with rate 2 per hour. Start from Oprn. (a) Find the probability that the second bus arrives after 1:30pm. (b) If there is no bus arriving between 1pm and 2pm, what is the expected number of buses arriving before 5pm? (c) If the waiting time T1 for the first bus is less than the inter-arrival time T2 between the first and second bus, what is the probability that the inter-arrival time T3 between the second and third bus is also less than T2? I (d) If exactly two buses have arrived during the hour from Opm to lpm, find the distribution function of the arrival time of the second bus and hence its expectation. (e) Suppose that passengers arrive at the station according to another in- dependent Poisson process with rate 6 per hour. All the passengers will wait until a bus arrives, and the bus takes away all the passengers