# Probability

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

1. 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 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.)

2. 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 per hour. Shuttle departures are independent of the process of passenger arrivals. Is the sequence of shuttle departures a Poisson process?

3. 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?

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

5. Find the PMF, pN(n) , of the number, N , of people on any given shuttle. Assume that lambda=20 and mu=2 .

For n>20 , pN(n)=

1. 👍 2
2. 👎 0
3. 👁 1,814
1. 2 . Yes it is a poisson process !!

1. 👍 8
2. 👎 0
2. 1) lambda
3) lambda + mu
4) 1/mu
Number 5 I don't have the answer

1. 👍 10
2. 👎 0
3. 5. 2*20^n/(22^(n+1))

1. 👍 11
2. 👎 0

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