Statistics 433 Practice Final Exam: Cover Sheet and Marking Sheet

Transcription

1 Statistics 433 Practice Final Exam: Cover Sheet and Marking Sheet YOUR NAME INSTRUCTIONS: No notes, no calculators, and no communications devices are permitted. Please keep all materials away from your desk. You if you need scratch paper use the back of the pages on the test. You should not have any blue books or stray papers near your desk. There are two blank pages at the end of this test that you can use as scratch paper. You should be attentive to clear presentation of your answers. You are encouraged to write as neatly as you can. You are also encouraged to use any extra time you have to check your work for accidental mistakes. The problems have equal weight, but, within a given problem, the parts typically have unequal weights. There may be some simple sort answer parts, and another parts may require a more substantial calculation. The parts receive weight based on the relative effort required for the part. You are expected to adhere strictly to the Academic Code of Ethics of the University of Pennsylvania. Leave the table below BLANK. Be sure to Print and Sign your name above. Problem Number Points Total

2 2 Problem 1. Consider a discrete time Markov chain {X n : n = 0, 1,...} with the state space S = {1, 2, 3, 4} and the transition matrix 1/4 1/4 1/4 1/4 p = 1/3 1/3 1/3 0 1/2 1/ For the first four parts, just circle True or False. Answer the other two parts in the space below. (1) This chain is irreducible. True or False? (2) This chain is aperiodic. True or False? (3) This chain is doubly stochastic. True or False? (4) This chain has an absorbing state. True or False? (5) Calculate P (X 3 = 1 X 0 = 4). (6) Let (π 1, π 2, π 3, π 4 ) be the stationary distribution of this chain. Give a clear and complete explanation why one has π 1 > π 2 > π 3 > π 4. Note: It would not be a good use of your time to compute the stationary distribution. This problem requires very little calculation.

3 3 Problem 2. Let N t, t 0 be a Poisson process with parameter λ > 0. (1) What is P (N t = 1)? (2) What is E[N t ]? (3) What is Var[N t ]? (4) What is the limiting value of t 1 P (N t 2) as t decreases to zero. (5) Calculate the value of E[N t (N t 1)] by any method you like.

4 4 Problem 3. Consider a continuous time Markov chain X t with state space 1, 2, 3 and with Q matrix given by Q = (1) If at time t 0 you are in state 1, what is the expect amount of time that you will remain there? (2) What is the stationary distribution of this continuous time chain? Justify your answer with a calculation. (3) What is the limiting value of P (X t = 1 X 0 = 1)?

5 5 Problem 4. Consider an M/M/ queueing system with arrival rate λ > 0 and service rate µ > 0. Let X t, t 0, be the number of people in the system at time t and assume X 0 = 0. (1) Suppose that λ > µ. What is the value of P (X t = k) as t? (2) Suppose that λ < µ. What is the value of P (X t = k) as t? (3) Suppose that there are 5 people in service at time t = 8, so X 8 = 5. If λ = 2 and µ = 3 what is the expected time until the number of people in service changes, i.e. goes up or goes down.

6 6 Problem 5. A European internet gaming site sometimes makes a lot of money and sometimes makes a REAL lot of money. Let s call these two states 1 and 2. Suppose that when the site is in state 1 the site makes A dollars per hour and when in state 2 it makes B dollars per hour (where A < B). If X t is the state of the site at time t we suppose that X t is a continuous time Markov chain with Q matrix given by Q = ( ) (1) Calculate the stationary distribution of this chain using the appropriate equation of stationarity. (2) What is your best estimate of the amount of money that the site makes over the time period [0, T ] for large values of T. Give your answer in terms of A and B. (3) Let p t (x, y) be the probability of being in state y at time t given that you are in state x at time zero. What is the limit of p t (1, 1) as t goes to infinity?

7 7 Problem 6. Consider a Markov chain with state space S = {0, 1, 2,...} and with transitions given by p(i, i + 1) = p for all i 0, p(i, i 1) = q for all i 1, and p(0, 0) = q. (1) If p > 1/2 is this chain transient, positive recurrent, or null recurrent? (2) If p < 1/2 is this chain transient, positive recurrent, or null recurrent? (3) If p = 1/2 is this chain transient, positive recurrent, or null recurrent? (4) Take p = 1/3. Either find the stationary probability distribution if it exists, or prove that one does not have a stationary probability distribution in this case.

8 8 Problem 7. You observe the outcomes of a sequence of flips of a fair coin. (1) If τ is the first time that you observe the pattern H, H, T, H what is the expected value of τ. (2) If τ is the first time that you observe the pattern H, T, H, H what is the expected value of τ. (3) Is the minimum of τ and τ a stopping time? Why, or why not? (4) Is the product of τ and τ a stopping time? Why or why not? (5) Is the difference τ τ a stopping time? Why or why not?

9 9 Problem 8.[Corrected 5/1/2017] There are two students Alice and Bob working at a fund raising car wash. Alice attracts passing cars at the rate of 3 per hour and she completes a car at a rate of 4 per hour. Bob does not do so well. He attracts cars a the rate of 2 per hour and completes a car at the rate of 3 per hour. They do their work and maintain their queues independently. (1) In the long run, what is the probability that Alice is working? (2) In the long run, what is the probability that Bob and Alice are both working at the same time? (3) In the long run, what is the expected number of cars waiting in Bob s queue?

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