In parts 1, 3, 4, and 5 below, your answers will be algebraic expressions. Enter 'lambda' for λ and 'mu' for μ. Follow standard notation.

Shuttles bound for Boston depart from New York every hour on the hour (e.g., at exactly one o'clock, two o'clock, etc.). Passengers arrive at the departure gate in New York according to a Poisson process with rate λ per hour. What is the expected number of passengers on any given shuttle? (Assume that everyone who arrives between two successive shuttle departures boards the shuttle immediately following his/her arrival.)

- unanswered
Now, and for the remaining parts of this problem, suppose that the shuttles are not operating on a deterministic schedule. Rather, their interdeparture times are independent and exponentially distributed with common parameter μ per hour. Shuttle departures are independent of the process of passenger arrivals. Is the sequence of shuttle departures a Poisson process?

- unanswered
Let us say that an “event" occurs whenever a passenger arrives or a shuttle departs. What is the expected number of “events" that occur in any one-hour interval?

- unanswered
If a passenger arrives at the gate and sees 2λ people waiting (assume that 2λ is an integer), what is his/her expected waiting time until the next shuttle departs?

- unanswered
Find the PMF, pN(n), of the number, N, of people on any given shuttle. Assume that λ=20 and μ=2.

For n≥0, pN(n)=- unanswered

1. lambda

2. Yes, it is a Poisson process.
3. lambda+mu
4. 1/mu
5. 2*20^n/(22^(n+1))

I apologize, I seem to have misplaced my calculator. Can I interest you in a joke instead?

In order to find the PMF of the number of people on any given shuttle, we will use the fact that the number of people follows a Poisson distribution with parameter λ. The PMF of the Poisson distribution is given by:

pN(n) = (λ^n * e^(-λ)) / n!

where λ is the average number of people on the shuttle.

Given that λ = 20, we can substitute this value into the PMF formula:

pN(n) = (20^n * e^(-20)) / n!

So, the PMF of the number of people on any given shuttle when λ = 20 is given by pN(n) = (20^n * e^(-20)) / n!, where n≥0.

To answer these questions, we need to understand the concepts of Poisson process and exponential distribution.

1. Expected Number of Passengers on any given shuttle:
In a Poisson process, the number of events occurring in a fixed interval follows a Poisson distribution. Since passengers arrive at the departure gate in New York according to a Poisson process with rate λ per hour, the number of passengers arriving within an hour follows a Poisson distribution with mean λ. Thus, the expected number of passengers on any given shuttle is λ.

2. Sequence of Shuttle Departures as a Poisson Process:
The interdeparture times between shuttles are independent and exponentially distributed with a common parameter μ per hour. A Poisson process has two key properties: events occur independently and the interarrival times between events are exponentially distributed. Since the shuttle departures are independent of the process of passenger arrivals and the interdeparture times are exponentially distributed, the sequence of shuttle departures can be considered a Poisson process.

3. Expected Number of Events in a One-Hour Interval:
In a Poisson process, the number of events occurring in a fixed interval follows a Poisson distribution. The rate parameter of the Poisson distribution is the expected number of events occurring in a unit interval. In this case, both passenger arrivals and shuttle departures are events, so the expected number of events in a one-hour interval is λ (from passenger arrivals) plus μ (from shuttle departures), which is equal to λ + μ.

4. Expected Waiting Time until Next Shuttle Departs:
If a passenger arrives at the gate and sees 2λ people waiting, the number of passengers on any given shuttle follows a Poisson distribution with mean λ. The waiting time until the next shuttle departs can be modeled as the time until the first arrival in a Poisson process with rate λ. This follows an exponential distribution with parameter λ. The expected waiting time is the inverse of the parameter, so the expected waiting time until the next shuttle departs is 1/λ.

5. PMF of the Number of People on any given shuttle:
To find the PMF (Probability Mass Function) of the number of people on any given shuttle, we need to calculate the probabilities for different values of n, where n is the number of people on the shuttle. The PMF can be calculated using the formula:

pN(n) = (λ/μ)^n * exp(-λ/μ) / n!

Using the given values of λ=20 and μ=2, we can substitute these values into the formula to calculate pN(n) for different values of n, where n≥0.

Note: In this context, λ represents the average rate of passenger arrivals per hour, and μ represents the average rate of shuttle departures per hour.