Boston University
Department of Electrical and Computer Engineering
EC505 STOCHASTIC PROCESSES
Norbert Wiener and Rudolf Kalman
Norbert Wiener Born: 1894, Columbia Missouri 1905: Entered Tufts college
CAS EC 505 Mathematics for Economics
Syllabus
Fall 2014
Course description
This is an introductory course in mathematics for economic analysis, aimed at MA students
with background in both economics a
Boston University
Department of Electrical and Computer Engineering
EC505 STOCHASTIC PROCESSES
Problem Set No. 3
Fall 2008 Issued: Wednesday, Sept. 17, 2008 Due: Friday, Sept. 26, 2008
Problem 3.1 Let
Boston University
Department of Electrical and Computer Engineering
EC505 STOCHASTIC PROCESSES
Notes on Notation
c 2010 W. C. Karl
Random Variables
X: Random variable
x: Value the vandom variable ta
EC 505 Mathematics for Economics
Assignment 4
Solution
1. Consider the function f : R2 ! R dened by f (x; y) = x2 ey :
(a) Draw a picture of the level sets.
(b) What is the slope of the level set at t
CAS EC 505 Mathematics for Economics
Practice Midterm Exam
Summer 2015
1. Consider the following matrix:
2
2
6 2
A=6
4 4
4
1
1
8
3
3
4
1 7
7:
7 5
9
(a) What is rank(A)? How many solutions does the sys
Boston University
Department of Electrical and Computer Engineering
EC505 STOCHASTIC PROCESSES
Problem Set No. 1
Fall 2008 Issued: Wednesday, Sept. 3, 2008 Due: Friday, Sept. 12, 2008
Problem 1.1 A ra
CAS EC 505
Review questions
Solutions
1. Exercise A1:3 (p 857).
Solution: a) x 2 (A \ B)c , x 2 A \ B , x 2 A and B , x 2 Ac or x 2 B c , x 2
=
=
c
c
A [B
2. Exercise 2:1 (p 15).
3. Exercise 2:21 (p 3
function [p,x,P]=pdf1d(xdata,xbins) % [p,x,P] = pdf1d(xdata,M) or [p,x] = pdf1d(ydata,xbins) % % xdata : Vector of data samples. % xbins : If xbins is a scalar, uses xbins equally spaced bins between
CAS EC 505 Mathematics for Economics
Final Exam
5/7/15
I Answer any three of the four questions
I Please write your answers in the blue books provided
I Please hand in the exam sheet with your blue bo
Boston University
Department of Electrical and Computer Engineering
EC505 STOCHASTIC PROCESSES
Problem Set No. 3 Solutions
Fall 2008 Issued: Wednesday, Sept. 17, 2008 Due: Friday, Sept. 26, 2008
Probl
CAS EC 505 Mathematics for Economics
Assignment 2
1. (a) Let f : Rn ! Rm and g : Rn ! Rm be two linear functions. Show that the function
f + g also is linear.
(b) Let h : Rn ! R be dened by h(x) = ax
Boston University
Department of Electrical and Computer Engineering
EC505 STOCHASTIC PROCESSES
Notes on Gaussian Variables, Vectors and Processes
c 2010 W. C. Karl
Gaussian Random Variables
Gaussian
Boston University
Department of Electrical and Computer Engineering
EC505 STOCHASTIC PROCESSES
Notes on Probability
c 2010 W. C. Karl
1. Probability Space
A triple (, F, P ) that describes outcomes o
Boston University
Department of Electrical and Computer Engineering
EC505 STOCHASTIC PROCESSES
Problem Set No. 1
Fall 2015
c 2015 D. A. Casta on
n
Issued: Wednesday, Sep. 2, 2015
Due: Friday, Sep. 11,
CAS EC 505 Assignment 2
Solution
1. (a) Let f : Rn ! Rm and g : Rn ! Rm be two linear functions. Show that the function
f + g also is linear.
(b) Let h : Rn ! R be dened by h(x) = ax + b: Show that h
CAS EC 505 Mathematics for Economics
Practice Final Exam
Summer 2015
1. Classify the stationary points for the following two functions.
(a) f : R3 ! R dened by:
f (x) = e
x1
x2 + 4x2 + 2x2 x3
2
+ 2x1
CAS EC 505 Mathematics for Economics
Syllabus
Summer 2015
Course description
This is an introductory course in mathematics for economic analysis, aimed at MA students
with background in both economics
CAS EC 505 Mathematics for Economics
Assignment 5
1. Solve the following optimization program:
max [ln x + ln y] subject to x2 + y 2 = 1
x;y
2. Find the solution to:
max [x + y] subject to x2 + y = 1
CAS EC 505 Mathematics for Economics
I. Review of Basic Concepts
Bjorn Persson
Boston University
Summer 2015
Bjorn Persson (Boston University)
Review
Summer 2015
1 / 43
Sets and Set Operations I
The c
CAS EC 505 Mathematics for Economics
II. Linear Algebra and Geometry
Bjorn Persson
Boston University
Summer 2015
Bjorn Persson (Boston University)
Linear Algebra
Summer 2015
1 / 83
Topics
Linear syste
CAS EC 505 Mathematics for Economics
III. Multivariate Calculus
Bjorn Persson
Boston University
Summer 2015
Bjorn Persson (Boston University)
Calculus
Summer 2015
1 / 38
Multivariate Functions I
Denit
CAS EC 505 Mathematics for Economics
IV. Optimization and Value Functions
Bjorn Persson
Boston University
Summer 2015
Bjorn Persson (Boston University)
Optimization
Summer 2015
1 / 79
Quadratic Forms
EC 505 Mathematics for Economics
Assignment 4
1. Consider the function f : R2 ! R dened by f (x; y) = x2 ey :
(a) Draw a picture of the level sets.
(b) What is the slope of the level set at the point
CAS EC 505 Mathematics for Economics
Assignment 3
1. (a) Are the following sets in R compact? Convex? Why/why not?
A
B
C
D
f1; 2; 3; 4; 5g
fx : 1 < x < 5g
A[B
A\B
=
=
=
=
(b) What about the following
CAS EC 505 Mathematics for Economics
IV. Optimization and Value Functions
Bjorn Persson
Boston University
Fall 2017
Bjorn Persson (Boston University)
Optimization
Fall 2017
1 / 82
Quadratic Forms I
De
Boston University
Department of Electrical and Computer Engineering
EC505 STOCHASTIC PROCESSES
Accounts on IMSIP
c 2014 W. C. Karl
1
SIGNET Lab
The Signals and Networks (SIGNET) LAB is a computational
CAS EC 505 Mathematics for Economics
L2 Vectors and Linear Independence
Bjorn Persson
Boston University
9/14/15
Bjorn Persson (Boston University)
Linear Algebra
9/14/15
1 / 25
Vectors
Denition
In Rn ,
CAS EC 505 Mathematics for Economics
III. Multivariate Calculus
Bjorn Persson
Boston University
Fall 2017
Bjorn Persson (Boston University)
Calculus
Fall 2017
1 / 50
Open and Closed Sets I
Denition of
CAS EC 505 Mathematics for Economics
Midterm Exam
3/1/17
Solution
1. (a) Rank(A) = 3. There are more rows than the rank, so there are at most one
solution
(b) x = (5; 1; 1)
(c) The n n matrix B is an
CAS EC 505 Mathematics for Economics
Assignment 3
1. (a) Are the following sets in R compact? Convex? Why/why not?
A
B
C
D
f1; 2; 3; 4; 5g
fx : 1 < x < 5g
A[B
A\B
=
=
=
=
(b) What about the following
CAS EC 505 Mathematics for Economics
Assignment 3
Solution
1. (a) Are the following sets in R compact? Convex? Why/why not?
A
B
C
D
f1; 2; 3; 4; 5g
fx : 1 < x < 5g
A[B
A\B
=
=
=
=
A is compact, but no
CAS EC 505 Mathematics for Economics
Midterm Exam
3/1/17
I Answer any three of the four questions
I Please write your answers in the blue books provided
I Please hand in the exam sheet with your blue