Homework #3, Beginning Markov Chains
1.
2.
3.
4.
The first three problems in Chapter 1, page 35 of the textbook.
2013 Final Exam, Problem 2, except use a computer to calculate the answer numerically.
2013 Exam 2, Problem 1.
The first four problems listed
Final Exam, Math 4058, Spring 2013
Problem 1. Ann, Beth and Courtney toss a ball around. They follow these rules:
Ann always throws the ball to Beth.
Beth is equally likely to toss the ball to either Ann or Courtney.
Courtney throws the ball to Ann 2/3
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Homework 1, Math 4058, Spring 2012
Problem 1. An insurance company examines its pool of auto insurance customers
and gathers the following information:
a) All customers insure at least one car.
b) 70% of the customers insure more than one car.
c) 20% of t
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Homework on Poisson Processes - Spring 2014
1) Let 2 have the standard normal distribution (Gaussian distribution) with mean 0 and variance 1.
Prove that the characteristic function for Z is g(t) = eA (-tAZ / 2). [Hintz One way to calculatate the
integral
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Homework 5, Math 4058, Spring 2014
Problem 1. Let Bt , t 0, be a standard Brownian motion (a Wiener process).
Calculate the following probabilities:
a) P(B4 > 1)
b) P(B4 > 1 | B2 = 1)
c) P(Bt > 1 for some time t 4)
Problem 2. Suppose you own one share of
30. Nancy reviews the interest rates each year for a 30-year ﬁxed mortgage issued on July 1.
She models interest rate behavior by a Markov model assuming:
(i) Interest rates always change between years.
(ii) The change in any given year is dependent on th
Final Exam, Math 4058, Spring 2014
20 points per problem for the rst 8 problems. 10 points per problem for the last 5
problems. Total points possible is 200 + 10 bonus points. Enjoy!
Problem 1. You are a medical claims processor for Humenscha Health Insur
Final Exam, Math 4058, Spring 2012
Problem 1. Let X and Y be two random variables with continuous probability
distributions.
a) Explain what the symbols E(X|Y = y) and E(X|Y ) are, explain generally
how one would calculate them from the joint distribution
Final Exam, Math 4058, Spring 2015
20 points per problem for the rst 8 problems. 10 points per problem for the last 5
problems. Total points possible is 200 + 10 bonus points. Enjoy!
Problem 1. Players A and B play a series of games that ends as soon as o
Homework #4, Ergodic Markov Chains
1.
2.
3.
4.
Textbook #5, Textbook #9
2013 Final Exam #7
2014 Final Exam #6
The Shoe Problem: Each morning, an individual leaves his house and goes for a run. He
is equally likely to leave either from his front or back do
Homework #6
1.
2.
3.
4.
5.
6.
7.
2014 Final Exam #1 (Poisson sampling theorem, Poisson reward process)
2014 Final Exam #3
SOA November 2001 Course 3 Exam, #30 (ask about this in class of youre stuck)
Textbook #3.2 (see theorem from class about sums of ind
Homework #5
Transient chains, random walks, and elementary Poisson processes
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2.
3.
4.
2015 Exam 1, #3
#1.8 of textbook
#1.14 of textbook
Let (X_n) be the symmetric random walk on the 2-dimensional integer lattice. Given that
you start at the origin, ca
Homework 1, Math 4058, Spring 2015
Problem 1. Let X and Y be continuous random variables with the joint probability
density function
f (x, y) =
x+y
0
if 0 x 1, 0 y 1,
otherwise
a) Find the conditional pdf of X, given that Y = 1/2.
b) Calculate E(X|Y = y)
Math 4058 Homework #2, Spring 2015
Due Monday Jan. 26.
1. Redo both parts of Problem #2 from the 2012 Final Exam (the deuce/tennis problem
that weve done in class), except replace the 0.6 probability that A wins a game with a
variable p, expressing your t
Homework #7
Conditional expectation, martingales and stopping times, Brownian motion
1.
2.
3.
4.
2013 Final Exam #6
Chapter 5 of text: #2,4, 7
Chapter 8 of text: #4, 9, 10, 11 (On #4d, look at Example 1 in Sec 8.2).
2014 Final Exam #8, 13.
5. A) Let B_t b
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Exam 2, Math 4058, Spring 2015
Problem 1. Siegbert runs a hot-dog stand in which customers arrive like a nonhomogeneous Poisson process. The instantaneous rate at which customers arrive
increases linearly from 0 at 8 am to a peak of 60 customers per hour
Exam 1, Math 4058, Spring 2015
Problem 1. Players A and B play a series of games that ends as soon as one side
has won two more games than the other side. On each individual game, Player A
wins with probability 2/3, independently of the outcomes of any ot
Exam 2, Math 4058, Spring 2013
Problem 1. A Markov chain Xn , n 0 with states 1, 2, 3 has the following transition
probability matrix:
1/2 1/3 1/6
0 1/3 2/3
1/2 0 1/2
If P(X0 = 1) = P(X0 = 2) = 1/4, calculate E(X2 ).
Problem 2. A diagram on the blackboa
36. The number of accidents follows a Poisson distribution with mean 12. Each accident
generates l, 2, or 3 claimants with probabilities }/ , )6 , )é , respectively.
Calculate the variance in the total number of claimants.
(A) 20
(B) 25
(C) 30
(D) 35
(E)