Name: _________________________
OR / STAT 645: Stochastic Processes
Final Exam
Dec. 16, 2010
You are bound by the GMU honor code. The exam is closed book, closed friends, etc. You may
use a calculator and 2 pages of notes (i.e., 4 sides of a page). Show all your work. Full credit
will not be given unless your steps are clear. Answer all questions. Box all answers. (I will
deduct points if you do not do this.)
Your final answers do not necessarily need to be numbers. However, you should simplify
formulas as much as possible.
1. (25 points) Customers arrive at a hotdog stand according to a Poisson process with rate  =
20 per hour. A customer buys 1 hotdog with probability 0.2, 2 hotdogs with probability 0.7,
and 3 hotdogs with probability 0.1.
a. What is the probability that no customers arrive during a half-hour period?
b. The vendor has 100 hotdogs. What is the approximate probability that the vendor
runs out of hotdogs during a two-hour period?
c. Suppose you know that 20 customers have arrived during the last hour. What is the
probability that 7 of those customers arrived during the last 15 minutes?
d. What is the probability the vendor sells exactly 3 hotdogs in a 5 minute period?
2. (25 points) Let X(t) = B(t) + t be Brownian motion with drift, where B(t) is standard
Brownian motion,  = 3, and  = 2. Assume that X(0) = 0.
a. Find P(X(5) > 13 | X(3) = 3).
b. Find P(X(3) > 3 | X(5) = 13).
c. Suppose that X(t) represents a stock price and that the stock pays a dividend every 2
time units. The dividend is $10 if the stock price has increased by more than $5 since
the last dividend; the dividend is $0 otherwise. What is the long-run rate of income
from dividends?
d. Extra credit (5 points): What is the answer to (c) if the dividends are paid at times
that occur according to a process with rate 1 per 2 time units.
3. (12 points) E-mails arrive in your inbox according to a Poisson process with rate 6 per hour.
You check your e-mail at intervals that are exponentially distributed with rate 2 per hour.
Each time you check your e-mail you delete 2 e-mails with probability 0.6 and 3 e-mails with
probability 0.4. (Assume your inbox goes to empty if you “delete” more e-mails than you
have.)
a. Give the rate transition matrix Q for this continuous-time Markov chain.
b. Is this Markov chain ergodic?
4. (8 points) In the electric power grid, the failure of one transmission line may cause failures of
other transmission lines, because the load carried on the failed line must be rerouted on other
lines. In turn, the new failures may cause even more failures, and so forth. This is known as a
cascading failure. Consider the following simplified model of failures in a power grid.
Suppose that when one transmission line fails, it causes 1 additional line to fail with
probability 0.6, 2 additional lines to fail with probability 0.3, and no additional lines to failure
with probability 0.1. This process repeats again for each newly failed power line. What is the
probability of a complete blackout – that is, what is the probability that the failures continue
indefinitely? (Assume an infinite number of transmission lines in the power grid.)
Final Exam
December 16, 2010
Name: _________________________
5. (5 points) Consider the following set of bets that you can make on the 2012 presidential race:
(1) A bet of $1 wins $2 if a democratic candidate wins, (2) a bet of $1 wins $2 if a republican
candidate wins, and (3) a bet of $1 wins $4 if an independent candidate wins. In each case,
you lose $1 if the appropriate candidate does not win. Suppose that you know that the
democratic candidate has a 90% chance of winning, the republican candidate has a 9%
chance of winning, and there is a 1% chance of an independent candidate winning. Find a set
of bets such that you are guaranteed to make money, regardless of the outcome.
6. (25 points) Consider the following discrete-time Markov chain and associated transition
probabilities. Suppose that you get a reward of 5 each time you are in state A, a reward of 7
each time you are in state B, and a reward of 9 each time you are in state C.
a. Using renewal theory, determine the long-run reward rate (per time step). (Start by
finding the expected time between visits to state A.)
b. Using the theory of Markov chains, determine the long-run reward rate (per time
step). (Start by solving for the steady-state probabilities j.)
0.4
0.2
A
B
0.8
0.6
1.0
C
GMU Honor Code Certification. I have abided by the GMU honor code in taking this
examination. The work on this exam is my own. I have received no assistance from other
persons during the exam.
_________________________________
Signature
Final Exam
______________
Date
December 16, 2010