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
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 ele
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.
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 a
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
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
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
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 h
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 o
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 th
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
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
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 functi
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 f
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 Higha
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 ot
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 
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 La
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 fun
Math 450  Homework 7
Solutions
1. Find the invariant distribution of the transition matrix:
0
1
0
1
2
.
P = 0
3
3
p 1p 0
The equation P = , and the normalization condition 1 + 2 + 3 = 1
give the syst
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
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 n
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
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.
The
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 a
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?
Solutio