10.6 Suppose that buses arrive at a particular stop according to a Poisson process with rate 6 per hour. I start waiting for the buss to arrive at 10:00.

a) What is the probability that no bus arrives in the next 20 min?

b) How many buses are expected to arrive in the next 90 min?

c) What is the variance of the number of buses during the next 3 h?

d) Somebody tells me that there has been no bus for one hour. Given this information, what is the probability that the next bus will arrive by 10:15?

e) What is the probability that no bus arrives in the next 15 min, given that I already have been waiting for 10 min for it.

f) At 10:15 I decide to go for a coffee. I return at 10:30 and see a bus leaving.

One of the people at the bus station tells me that this was the second since I left the station and there has been another bus during my leaving the station.

What is the probability that the first bus I did not even see arrived at the bus station within 1 min after I left (10:15 to 10:16)