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
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 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
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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
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)
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
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
LEVEL 1
Sheila walks to the Dead Sea and throws a rock into the lake. Since the Sea is calm, ripples in the shape of
concentric circles are formed on the water. If the radius of the outer ripple is increasing at a rate of 2 feet per
second, at what rate i
Solutions to Worksheet for Lesson 19 (Section 4.1)
Related Rates
Math 1a
November 7, 2007
1. A 10 ft ladder leans against the side of a building. If the top of the ladder begins to slide down
the wall at the rate of 2 ft/sec, how fast is the bottom of the
1. Explain how to find 30 80 using mental math.
To find 30 X 80, you _
_
_.
2. Mr. Burns class is taking a field trip to the planetarium. The trip will
cost $27 for each student. There are 18 students in his class.
a. Round each factor (number being multi
Related Rate Problems
Problem 1: (Depth) A conical (cone-shaped) tank (with vertex down) is 10 feet across the top
and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the
rate of change of the depth of the water
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 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
1.
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