You arrive at a bus stop where buses come at a rate of $3$ per hour. What's the probability distribution of the waiting time for the next bus and its mean if the interarrival times between buses are
(a) constant,
(b) exponential,
(c) either 0 or 60 minutes (groups of 3 buses go by in an hour).
I can guess that the mean waiting time for constant is just $c/2$ where $c$ is the constant. In exponential distribution it will be just $\lambda$. Not so sure about the third one.
But getting these means doesn't help me find the distribution. Can anyone suggest me how to do it? I think you need to use the equilibrium distribution, but where do we use the fact that buses come at $3$ per hour? Do I even need this for the constant case?
Assume without loss of generality that the buses in each group of $3$ are ordered. The probability of the next bus being the second or third is zero, since there is no gap between the group's buses. Effectively they may be treated as a single bus that comes once an hour, whereupon the expected waiting time is half an hour.