Math 350 - Homework 1 - Solutions
1. Interpret the following Matlab expressions:
(a) r = rand; a = (r <= 1/2)
The random variable a takes only the values 1 or 0. It is 1 if and only if r is less than or equal to
1/2, so a = 1 with probability 1/2 and 0 wi
Math 350 Fall 2012 - Homework 5
Due 10/05/2012
1. Read text, Chapter 3, pages 101 to 107. (There are interesting topics I have skipped from Chapter
2; I plan to return to them later.)
2. Approximate Brownian motion on the plane. We dene a stochastic proce
Math 350 Fall 2012 - Homework 8
Due 11/9/2012
Here are some general geometric denitions and background facts related to the problems below.
(All but the last problem are to be solved by hand.) The n-dimensional ball B n (r) of radius r is the
subset of Rn
Math 350 Fall 2012 - Homework 10
Due 11/30/2012
The Ising model. The Ising model originated in statistical physics as a mathematical model for ferromagnetism,
but it has found much broader applicability, in addition to having great theoretical interest, a
Math 350 Fall 2012 - Homework 9
Due 11/19/2012
The hard-core model. (This example is adapted from Finite Markov Chains and Algorithmic Applications, by
Olle Hggstrm, London Mathematical Society, Student Texts 52, Cambridge University Press, 2002.) The har
Math 350 Fall 2012 - Homework 6
Due 10/12/2012
Denitions for problem 1. Problem 1 deals with the class structure of a Markov chain. It is sometimes
possible to break a Markov chain into smaller pieces that can be analyzed separately from the whole, based
Math 350 Fall 2012 - Homework 11
Due 12/07/2012
Simulated annealing and the traveling salesman problem. See pages 139-146 of textbook. The traveling salesman
problem is a widely studied model optimization problem. We are given m + 1 cities, indexed by the
Math 350 Fall 2012 - Homework 7
Due date: 10/29/2012
The beta distribution. The beta distribution is a family of continuous probability distributions dened on the
interval [0, 1], parametrized by two positive parameters, , , having probability density fun
Math 350 - Homework 2
Due 2/05/2010
1. (Text, problem 7, page 35.) If X and Y have a joint probability density function given by
f (x, y) = 2e(x+2y)
for x and y in (0, ), nd the probability P (X < Y ).
2. (Text, problem 9, page 36.) The continuous random
Math 350 - Homework 3
Due 2/12/2010
1. (Text, problem 25, page 37.) The bus will arrive at a time that is uniformly distributed between 8 and
8 : 30 A.M. If we arrive at 8 A.M., what is the probability that we will wait between 5 and 15 minutes?
2. (Text,
Math 350 - Homework 1
Due 1/29/2010
The probabilistic experiment consisting of picking a random number between 0 and 1 with the uniform
probability distribution over the interval [0, 1] is approximately realized on a computer by generating a pseudorandom
Math 350 - Homework 4
Due 2/19/2010
1. (Text, problem1, page 46.) If x0 = 5 and
xn = 3xn1 (mod 150)
nd x1 , . . . , x10 .
2. (Text, problem 6, page 47.) This problem refers to the integral
x(1 + x2 )2 dx.
I=
0
(a) Find the exact value of I.
(b) Estimate t
Math 350 - Midterm test - Solutions
1. If X and Y have a joint probability density function given by
f (x, y ) = 2e(x+2y)
for x and y in (0, ), nd the probability P (X < Y ).
y
2e(x+2y) dx dy
P (X < Y ) =
0
0
y
2y
=2
ex dx dy
e
0
0
=2
0
e2y (1 ey )dy
e2y
Math 350 - Homework 5
Due 2/26/2010
1. (Text, problem 4, page 63.) A deck of 100 cardsnumbered 1, 2, . . . , 100is shued (i.e., a random
permutation is applied to the cards in the deck) and then turned over one card at a time. Say that a
hit occurs whenev
Math 350 - Homework 4
Due 9/28/2012
1. Read text, chapter 2, pages 51 - 69.
2. (Problem 8, chapter 2, page 95) The sphere S n1 is dened to be the hypersurface in Rn consisting
of points x = (x1 , . . . , xn ) such that x2 + + x2 = 1. Note that S 1 is the
Math 350 Fall 2012 - Homework 3
Due 9/21/2012
1. Browse the Notes on Random Variables:
http:/www.math.wustl.edu/~feres/Math350Fall2012/Notes_on_random_variables.pdf
(You can nd a link to it on the Course Plan part of the online syllabus.) Some of the mate
Math 350 - Homework 4
Solutions
1. Read text, chapter 2, pages 51 - 69.
2. (Problem 8, chapter 2, page 95) The sphere S n1 is dened to be the hypersurface in Rn consisting
of points x = (x1 , . . . , xn ) such that x2 + + x2 = 1. Note that S 1 is the circ
Math 350 Fall 2012 - Homework 3
Solutions
1. Browse the Notes on Random Variables:
http:/www.math.wustl.edu/~feres/Math350Fall2012/Notes_on_random_variables.pdf
(You can nd a link to it on the Course Plan part of the online syllabus.) Some of the material
Math 350 - Homework 2 - Solutions
1. Read text, section 1.2 (pages 10 - 27.)
2. (Text, problem 8, page 44) Random walk with drift. Use a biased coin to simulate a random walk of
30 steps on the line. Begin at x = 0. If the coin falls heads (H), take one s
Math 350 - Homework 1 - Solutions
1. Interpret the following Matlab expressions:
(a) r = rand; a = (r <= 1/2)
The random variable a takes only the values 1 or 0. It is 1 if and only if r is less than or equal to
1/2, so a = 1 with probability 1/2 and 0 wi
Math 350 Fall 2012 - Homework 5
Solutions
1. Read text, Chapter 3, pages 101 to 107. (There are interesting topics I have skipped from Chapter
2; I plan to return to them later.)
2. Approximate Brownian motion on the plane. We dene a stochastic process X
Math 350 Fall 2012 - Homework 6
Solutions
Denitions for problem 1. Problem 1 deals with the class structure of a Markov chain. It is sometimes
possible to break a Markov chain into smaller pieces that can be analyzed separately from the whole, based on
th
Math 350 Fall 2012 - Homework 10
Solutions
The Ising model. The Ising model originated in statistical physics as a mathematical model for ferromagnetism,
but it has found much broader applicability, in addition to having great theoretical interest, as a q
Math 350 Fall 2012 - Homework 9
Solutions
The hard-core model. (This example is adapted from Finite Markov Chains and Algorithmic Applications, by
Olle Hggstrm, London Mathematical Society, Student Texts 52, Cambridge University Press, 2002.) The hard-cor
Math 350 Fall 2012 - Homework 8
Solutions
Here are some general geometric denitions and background facts related to the problems below.
(All but the last problem are to be solved by hand.) The n-dimensional ball B n (r) of radius r is the
subset of Rn den
Math 350 Fall 2012 - Homework 7
Solutions
The beta distribution. The beta distribution is a family of continuous probability distributions dened on the
interval [0, 1], parametrized by two positive parameters, , , having probability density function
f (x
Math 350 Fall 2012 - Homework 11
Solutions
Simulated annealing and the traveling salesman problem. See pages 139-146 of textbook. The traveling salesman
problem is a widely studied model optimization problem. We are given m + 1 cities, indexed by the set
Math 350 - Homework 1
Due 9/07/2012
The probabilistic experiment consisting of picking a random number between 0 and 1 with the uniform
probability distribution over the interval [0, 1] is approximately realized on a computer by generating a pseudorandom
Math 350 - Homework 2
Due 9/14/2012
1. Read text, section 1.2 (pages 10 - 27.)
2. (Text, problem 8, page 44) Random walk with drift. Use a biased coin to simulate a random walk of
30 steps on the line. Begin at x = 0. If the coin falls heads (H), take one
Math 350 - Midterm test - March 5, 2010
1. If X and Y have a joint probability density function given by
f (x, y) = 2e(x+2y)
for x and y in (0, ), nd the probability P (X < Y ).
2. Let X be a binomial random variable X with parameters (n, p). Explain why