Poisson Distribution Help

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
hour.

(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.

??

not quite sure if im going in the right direction
a) lamda
b) μt * e^(-μt)
c)
λ/(λ+μ) x λ + μ/(λ+μ) x μ

d)
E = 1/2λ + 1/λ

e)
μt * e^(-μt) + λt * e^(-λt)

mussi said:

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
hour.

(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.

??

Click to expand...

Thank you for showing more of your work.

a) lambda....Yes - has to average out over time.
b) μt * e^(-μt)....NO factor of t, just:... μ e^(-μt)
c)
λ/(λ+μ) x λ + μ/(λ+μ) x μ....TOO MUCH WORK

d)
E = 1/2λ + 1/λ

e)
μt * e^(-μt) + λt * e^(-λt)

Click to expand...

c) The two kinds of "event" are completely independent of each other. The sum is just the sum:
events per hour = lambda + µ

d) Shuttle departures are again independent of passengers, and also each departures is an independent event. If you start a clock when the passenger arrives, what is the expectation value of the exponential distribution?

e) This part needs more thought. I suspect you will have to convolute the two distributions by integrating the product. I am quite sure it will not be the simple sum you have shown (BTW, remember there is no factor of t in the exponential). Show us more of your work on this part - in the meantime I will ponder it a bit also.

HINT: the expectation value of passengers per shuttle = (passengers/hour)/(shuttles/hour) = lambda / µ

Another HINT: the form of the Poisson distribution for integer number \(\displaystyle x\) of discrete events in time \(\displaystyle t\) is

\(\displaystyle \displaystyle P(x;\ t) = \dfrac{\mathrm e^{-\lambda\ t}(\lambda\ t)^x}{x!}\)