SYS 6005 – Assignment
Instructions
• Solve the following exercises.
• No late submission will be allowed.
• Give quantitative answers where required; must explain your reasoning.
• The total point for this exam is 100.
• CLEAR WRITING.
Question 1 (total 40 points)
This question is based on a time when people needed to log in to email servers to write emails. Assume that Adam logs in to an email server at time zero. He then spends his time exclusively on writing emails. The times that his emails are sent can be modeled by a Poisson process with a rate λA/hour. His friend Brianna also logs in to the server at time 1 (which means 1 hour later) and starts typing emails independently of Adam. The time that her emails are sent also can be modeled by a Poisson process with a rate λB/hour.
(a) (6 points) Let Y1, and Y2 be the times at which Adam’s first and second emails were sent. Find
E[Y2|Y1].
(b) (8 points) Similar to the last question, let Y1 and Y2 be the times at which Adam’s first and second emails were sent. When Brianna logs in to the server, you are told that Adam has sent exactly one email so far. (b.1) (4 points) What is the conditional expectation of Y1 given this information? (b.2) (4 points) What is the conditional expectation of Y2 given this information?
(c) (8 points) Find out the PMF of the total number of emails sent by two of them together duringthe interval of [0,2] (which means in the first two hours).
(d) (8 points) Given that a total of 10 emails were sent during the period of [0, 2] (the first two hours),what is the probability that exactly four emails were sent by Adam?
(e) (10 points) Suppose that λA = 4. Use Chebyshev’s (4 points) and Markov’s (4 points) inequality to find upper bounds on the probability that Adam sent at least five emails during the time interval [0, 1]. Which one provides a better bound (2 points)?
Question 2 (total 20 points)
The country of Northeros is divided into m distinct regions based on land geology, which determines earthquake probability. During any given hour, the probability of an earthquake in the region i ∈ {1,…,m} is pi ≈ 0.001i, and is independent of all other hours. Find a good approximation for the probability that there are exactly 10 earthquakes over a period of 10000 hours in the entire Northeros. You may assume that: (a) earthquakes in different regions are independent, and (b) there is at most one earthquake during a one-hour period in any given region. Note that there can be multiple earthquakes in the entire country during a one-hour period.
Question 3 (total 35 points)
Aber and Byft are two competing mobile ride-sharing companies serving one million customers jointly. Suppose that each customer chooses from Aber and Byft according to the following rules:
• Each customer can only choose one of the two companies in each month, and must stay with that company for the entire month, before considering switching the next month.
• A new customer of Aber is someone who was not using Aber in the prior month. Such a customer is equally likely to stay with Aber (a loyal customer) or to switch (become a new customer to Byft) next month. The same holds true for a new customer of Byft.
• A loyal customer is someone who used the same company in the prior month. Such a customer has 3/4 probability of staying loyal and a 1/4 probability of switching to the competitor.
As shown in the following figure, a Markov chain can be used to model the usage behavior of customers in the above setting. Here, AL stands for loyal customers of Aber, AN stands for new customers to Aber, BL stands for loyal customers of Byft, BN stands for new customers to Byft.
(a) (7 points) At a given time, suppose that the system has run for a long time and is stabilized.What fraction of the one million users should one expect to be new customers of Aber?
(b) (7 points) For a loyal customer of Aber, what is the expected duration in months, that s/he staysas a loyal customer of Aber before switching?
(c) (7 points) For a loyal customer to Aber, what is the expected time in months, until s/he becomesa loyal customer to Byft?
(d) (7 points) After the system was stabilized, the CEO of Aber came up with a marketing idea tooffer new customers of Aber a cash bonus of $10. How much should the CEO expect to pay for such cash bonuses in the first month of implementing the policy?
(e) (7 points) With the cash-bonus policy implemented for a long time, Aber observes that more newcustomers are now becoming loyal to them (3/4 of them as opposed to 1/2 before the offer). To keep this going, how much Aber needs to pay on average on cash-bonus per month?
Question 4 (total 10 points)
For each of the following statements, explain whether it is true or false. If you think it is true, then prove why you think it is always true. If you believe it is false, then giving a counterexample is sufficient.
(a) (5 points) If the posterior distribution is symmetric, then the MAP and LMS estimators coincide. (b) (5 points) Let X and Y be two discrete random variables. If P(X > Y ) > 0.5, then E[X] > E[Y ].
