Ma 450: Mathematics for Multimedia Solution: to Homework Assignment 1
Prof. Wickerhauser Due Friday, February 5th, 2010
1. Suppose a divides 2b and 2b divides c. Must a divide c? Solution: Yes. Rename B = 2b and apply the transitive property of divisibili
Notes for Math 450
Lecture Notes 1
Renato Feres
1
Probability Models
We often have to deal with situations involving hidden or unpredictable factors.
Sometimes the unpredictability is an essential element of the phenomenon under investigation. For example
Notes for Math 450
Lecture Notes 2
Renato Feres
1
Probability Spaces
We rst explain the basic concept of a probability space, (, F , P ). This may be
interpreted as an experiment with random outcomes. The set is the collection
of all possible outcomes of
Notes for Math 450
Lecture Notes 3
Renato Feres
1
Moments of Random Variables
We introduce some of the standard parameters associated to a random variable.
The two most common are the expected value and the variance. These are
special cases of moments of
Math 450  Homework 2
Due date: Friday, 2/02/07
1. Read all of Lecture Notes 02
2. Do Exercise 4.1 of LN02(Show your program)
3. Do Exercise 4.2 of LN02
4. Do Exercise 4.5 of LN02(Show your program)
5. Do Exercise 4.11 of LN02
6. Do Exercise 4.14 of LN02
Math 450  Homework 1
Due date: Friday, 1/26/07
1. Read all of Lecture Notes 01.
2. Do Exercise 2.2 of LN01(Show your program.)
3. Do Exercise 2.3 of LN01
4. Do Exercise 2.4 of LN01
5. Do Exercise 2.6 of LN01 (Show your program.)
6. Do Exercise 2.8 of LN0
Math 450  Homework 3
Due date: Friday, 2/09/07
1. Read all of Lecture Notes 03. We wont be using all of it now, but it will
be helpful if you try to familiarize yourself with what is in there.
2. Do Exercise 6.2 of LN03.
3. Do Exercise 6.4 of LN03.
4. Do
Math 450  Homework 4
Due date: Friday, 2/16/07
1. Read the introduction and sections 1.1, 1.2, 1.3 and 1.4 of the rst chapter
of Norris book.
2. Go back to our samplefrommarkov program, think about how it works,
then read Exercise 1.1.3, page 8, of Norri
Math 450  Homework 5
Due date: Friday, 2/23/07
1. Read sections 1.5, 1.6 and 1.7 of the textbook. (Also review sections 1.3
and 1.4, which are needed for problems 2, 3, and 4.)
2. Do Exercise 1.3.2 of the textbook. In addition to answering the questions
Topics in Applied Mathematics Mathematics for Multimedia
MATH 450

Spring 2014
Ma 450: Mathematics for Multimedia
Solution:
to Homework Assignment 4
Prof. Wickerhauser
Due Friday, March 28th, 2014
1. Use the divided dierences method to t a polynomial, in Newtons form, through the points (1, 3),
(2, 4), (4, 5), and (9, 4). Evaluate i
Topics in Applied Mathematics Mathematics for Multimedia
MATH 450

Spring 2014
Ma 450: Mathematics for Multimedia
Solution:
to Homework Assignment 6
Prof. Wickerhauser
Due Friday, April 25th, 2014
1. Fix an integer q > 0, let N = 2q > 1 and consider a graph with vertices labeled 0, 1, . . . , N 1.
Suppose that vertex i is connected
Topics in Applied Mathematics Mathematics for Multimedia
MATH 450

Spring 2014
Ma 450: Mathematics for Multimedia
Solution:
to Homework Assignment 2
Prof. Wickerhauser
Due Friday, February 14th, 2014
1. Let N be a xed positive integer.
(a) How many vertices are there in the unit cube in Euclidean N space?
(b) Fix a vertex in the N
Topics in Applied Mathematics Mathematics for Multimedia
MATH 450

Spring 2014
Ma 450: Mathematics for Multimedia
Solution: to Homework Assignment 5
Prof. Wickerhauser
Due Friday, April 11th, 2014
t
1. Draw the graphs of w( 3 4) and w( t4 ) on one set of axes for the Haar function w(t)
3
dened in Equation 5.2.
The graphs are shown i
Topics in Applied Mathematics Mathematics for Multimedia
MATH 450

Spring 2014
Ma 450: Mathematics for Multimedia
Solution:
to Homework Assignment 3
Prof. Wickerhauser
Due Friday, February 28th, 2014
1. Suppose that f (t) = 0 if t 1, and 1 < f (t) < 2 for all t (1, 1). Let f1 be the 1periodization of
f.
(a) Prove that f1 (t)
Math 450  Homework 9
Solutions
1. Exercise 3.3.1, page 114. Consider the Markov chain on cfw_1, 2, 3, 4 with
generator matrix
1
1/2
1/2
0
1/4 1/2
0 1/4
.
1/6
0 1/3 1/6
0
0
0
0
Calculate (a) the probability of hitting 3 starting from 1, and (b) the
ex
Math 450  Homework 11
Solutions
1. We x a time T and wish to observe the distribution of positions of WT ,
the onedimensional Brownian motion. Do this as follows. Using the program bpath2.m in Highams paper, simulate a large number of sample
paths of Wt
Ma 450: Mathematics for Multimedia Solution: to Homework Assignment 2
Prof. Wickerhauser Due Friday, February 19th, 2010
1. Fix a vertex in the N cube, the unit cube in Euclidean N space. How may other vertices are connected to it by single edges? Solut
Ma 450: Mathematics for Multimedia Solution: to Homework Assignment 3
Prof. Wickerhauser Due Friday, March 5th, 2010
1. Suppose that f (t) = 0 if t 1, and f (t) < 1 for all t (1, 1). Prove that f1 (t) < 2 for all t R, where f1 is the 1periodizatio
Ma 450: Mathematics for Multimedia Solution: to Homework Assignment 4
Prof. Wickerhauser Due Friday, April 2nd, 2010
1. Find the Lagrange polynomial through the points (1, 3), (4, 3). Solution: The Lagrange polynomial through the points (1, 3), (4, 3) is
Ma 450: Mathematics for Multimedia Solution: to Homework Assignment 5
Prof. Wickerhauser Due Friday, April 16th, 2010
t 1. Draw the graphs of w( 3  4) and w( t4 ) on one set of axes for the Haar function w(t) defined 3 in Equation 5.2.
Solution:
The gra
Math 450  Homework 6
Solutions
1. Exercise 1.7.1. Find all the invariant distributions of the transition matrix
1
0001
2
2
0 1 0 1 0
2
2
P = 0 0 1 0 0
0 1 1 1 1
4
4
4
4
1
0001
2
2
We need to solve the system = P with the extra normalization condition i
Math 450  Homework 3
Solutions
1. Exercise 6.2, LN03. Let X be a random variable with mean m and
standard deviation . Chebyshevs inequality reads:
1
.
K2
Therefore, the probability that X lies on the interval (m 5, m + 5 ) is
P (X m K )
P (X m < 5 )
Math 450  Homework 2  Solutions
(Exercises from Lecture Notes 2)
1. Exercise 4.1 To count the number of ways to get a 9, Ill count how
many triples (n1 , n2 , n3 ) of numbers from 1 to 6 such that n1 n2 n3
and n1 + n2 + n3 = 9, and then the number of pe
Math 450  Homework 1  Solutions
(Exercises from Lecture Notes 1)
1. Exercise 2.2
300
250
200
150
100
50
0
0
1
2
3
4
5
6
7
8
9
10
Figure 1: Histogram for 1000 runs of the experiment: toss a coin 10 times and
count the number of heads.
%
u=121; eed for th
Math 450  Homework 5
Solutions
1. Exercise 1.3.2, textbook. The stochastic matrix for the gambler problem has the following form, where the states are ordered as (0, 2, 4, 6, 8, 10):
1
1
2
1
P = 2
0
0
0
0
0
0
1
2
0
0
0
0
0
0
0
1
2
0
0
0
0
1
2
0
0
0
1
2
0
Math 450  Homework 8
Solutions
1. Given the generator matrix
2
1
1
0
Q = 4 4
2
1 3
we look for p11 (t) of P (t) = etQ . The eigenvalues of Q are the roots of
det(I Q) = ( + 4)( + 2)( + 3) 6) = 0.
They are = 0, 4, 5. Therefore, it is possible to nd an inv
Math 450  Homework 10
Solutions
1. Consider a eet of N buses. Each bus breaks down independently at rate
and is then sent to the depot for repair. The repair shop can only repair
one bus at a time and each bus takes an exponential time of parameter
to
Topics in Applied Mathematics Mathematics for Multimedia
MATH 450

Spring 2014
Ma 450: Mathematics for Multimedia
Solution:
to Homework Assignment 1
Prof. Wickerhauser
Due Friday, January 31st, 2014
1. Suppose a divides b and b divides c and c divides d. Must a divide d?
Solution:
Yes. Apply the transitive property of divisibility t