# 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.)

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?

Yes, it is a Poisson process.

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?

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?

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)=

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## 1. lambda

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

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## Correct ans for (5)

5. (2/22)*(20/22)^n

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## To answer these questions, we need to understand some concepts and formulas related to Poisson processes.

A Poisson process is a stochastic process that models the occurrence of events over time. It has the following properties:
1. The number of events that occur in any given interval is independent of the number of events in any disjoint interval.
2. The probability of a single event occurring in a small time interval is proportional to the length of the interval.
3. The probability of multiple events occurring in a small time interval is negligible.

Now, let's tackle the individual questions:

1. Expected number of passengers on any given shuttle:
In a Poisson process, the expected number of events in a given interval is equal to the product of the rate (λ) and the length of the interval. In this case, the interval is one hour, so the expected number of passengers on any given shuttle is λ.

2. Sequence of shuttle departures:
Since the interdeparture times are exponentially distributed with parameter μ, the sequence of shuttle departures is not a Poisson process. In a Poisson process, the interarrival times between events (in this case, shuttle departures) should be exponentially distributed, not the interdeparture times.

3. Expected number of events in one-hour interval:
The expected number of events in a Poisson process can be calculated using the formula λ*T, where λ is the average rate of events per time unit and T is the length of the interval. In this case, λ is still the rate of passenger arrivals per hour, so the expected number of events in a one-hour interval is λ.

4. Expected waiting time until the next shuttle departs:
If a passenger arrives and sees 2λ people waiting, it means that there are already 2λ passengers in the queue. Since the shuttle departures are independent of passenger arrivals, the waiting time until the next shuttle departs is the same as the waiting time for the 2λth passenger in a queue with a departure rate of μ per hour. The expected waiting time for the 2λth passenger can be calculated using the formula (2λ)/(μ).

5. PMF of the number of people on any given shuttle:
To find the PMF of the number of people (N) on a shuttle when λ=20 and μ=2, we can use the formula:
pN(n) = (e^(-μ) * μ^k) / k!, where k is the number of people on the shuttle and e is the base of the natural logarithm. Substitute the values of λ and μ into the formula to get the PMF for N.