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MATH 303 Study Guide - Midterm Guide: Queueing Theory, Urn Problem, Poisson Point ProcessExam


Department
Mathematics
Course Code
MATH 303
Professor
all
Study Guide
Midterm

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The University of British Columbia
Sessional Exams – 2010 Term 2
Mathematics 303 Introduction to Stochastic Processes
Dr. G. Slade
Last Name: First Name:
Student Number:
This exam consists of 8questions worth 10 marks each. No aids are permitted.
Please show all work and calculations. Numerical answers need not be simplified.
Problem total possible score
1. 10
2. 10
3. 10
4. 10
5. 10
6. 10
7. 10
8. 10
total 80
1. Each candidate should be prepared to produce his library/AMS card upon request.
2. Read and observe the following rules:
No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave
during the first half hour of the examination.
Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities
in examination questions.
CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the
examination and shall be liable to disciplinary action.
(a) Making use of any books, papers or memoranda, other than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness
shall not be received.
3. Smoking is not permitted during examinations.

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April 27 2010 Math 303 Final Exam Page 2 of 9
1.(10 points) You are managing a vending machine which generates an average daily profit of $2000 when
it is working, but it breaks down each day independently with probability q= 0.2. A repair
company offers a service contract to restore the machine to working order with probability p
in one day whenever it breaks down, at a cost of $ 100
1peach day (whether the machine is
working or not). You get to choose p. What value of pshould you choose in order to
maximize your expected profit?

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April 27 2010 Math 303 Final Exam Page 3 of 9
2. The two-parameter Ehrenfest urn model is defined as follows. A total of Mballs are divided
between two urns, urn Aand urn B. A ball is chosen uniformly at random. If it is chosen
from urn Athen it is placed in urn Bwith probability b, and otherwise it is returned to urn
A. Similarly, if it is chosen from urn Bthen it is placed in urn Awith probability a, and
otherwise it is returned to urn B. Let Xnbe the number of balls in urn Aat time n.
(a)(3 points) Determine the transition probabilities for this Markov chain.
(b)(4 points) Guess (or otherwise determine) the stationary distribution for this Markov chain.
(c)(3 points) Verify that your guess is correct and that the Markov chain is reversible.
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