Hi, I'm having difficulty in these questions. Appreciate The help!

(a) Shuttles depart from New York to Boston every hour on the hour. Passengers
arrive according to a Poisson process of rate λ per hour. Find the expected
number of passengers on a shuttle. (Ignore issues of limited seating.)

Simply λ?

(b) Now, and for the rest of this problem, suppose that the shuttles are not operating
on a deterministic schedule, but rather their interdeparture times are
exponentially distributed with rate μ per hour, and independent of the process
of passenger arrivals. Find the PMF of the number shuttle departures in one

(μt * e^(-μt) )/0!) ?

(c) Let us define an “event” in the airport to be either the arrival of a passenger, or
the departure of a plane. Find the expected number of “events” that occur in
one hour.


(d) If a passenger arrives at the gate, and sees 2λ people waiting, find his/her expected
time to wait until the next shuttle.


(e) Find the PMF of the number of people on a shuttle.