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 (age 11) 1909: Harvard graduate student in zoology (ag
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 and mathematics. The course consists of three parts.
In
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 Z be a random variable with the following exponential
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 book
1. (a) Let f : Rn ! R and g : Rn ! R2 be smooth func
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 the point (2; 0)? What about the point (0; 0)?
(c) Speci
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 random experiment consists of tossing a die and observing
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 system Ax = b have?
(b) Find the solution(s) to the system
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 34).
Solution: a) f is innitely many times dierentiable
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 % min(xdata) and max(xdata). If xbins is a vector, uses
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
Problem 3.1 Let Z be a random variable with the following ex
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 takes on
PX (x): = Pr(X x) Probability distribution func
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 probability density function:
pX (x) =
1
(x mx )2
exp
EC505
STOCHASTIC PROCESSES
Class Notes
c 2015 Prof. D. Castaon & Prof. W. Clem Karl
n
Dept. of Electrical and Computer Engineering
Boston University
College of Engineering
8 St. Marys Street
Boston, MA 02215
Fall 2015
2
Contents
1 Introduction to Probabil
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 lab available to you as a student in EC505
for complet
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 I
Denition
A quadratic form is a function f : Rn ! R th
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 of a random experiment
(a) : Set of elementary outcomes,
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
3x2 + 8x3 :
3
(b) g : R3 ! R dened by g(x) = xT Ax wher
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 and mathematics. The course consists of ve parts. In
t
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
x;y
Also nd the equation of the tangent to the level cu
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 (2; 0)? What about the point (0; 0)?
(c) Specify the di
CAS EC 505 Mathematics for Economics
Assignment 5
1. Suppose f : R2 ! R is dened by:
x4 + a(x2
f (x; a) =
1)
where x 2 X = [ 1; 1] and where a 2 R:
(a) Find the solution correspondence x(a) for all a 2 R: Draw a graph.
(b) Find and draw a graph of the the
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 + b: Show that h is not linear.
2. (a) Express x = (1;
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 sets in R2 ?
E = [0; 1] f1g
F = fx : kx rk
G = \n2N 0;
Boston University
Department of Electrical and Computer Engineering
EC505 STOCHASTIC PROCESSES
Notes on LTI Input/Output Relationships for Single-Input/Single-Output Systems
c 2010 W. C. Karl
Deterministic Case
N (J)
D (J)
O (J)
Signals
General Systems
Li
CAS EC 505 Mathematics for Economics
L3 The Determinant
Bjorn Persson
Boston University
9/16/15
Bjorn Persson (Boston University)
Linear Algebra
9/16/15
1 / 18
The Determinant I
The determinant
Consider the linear transformation f : Rn ! Rn represented by
CAS EC 505 Mathematics for Economics
Midterm Exam
8/14/15
I Please write your answers in the blue booklets provided
I Write your name on the blue book in pen
I Good luck!
1. Let x = (2; 2; 1) and y = (1; 0; y3 ):
(a) Find y3 such that the angle between x
CAS EC 505 Mathematics for Economics
Assignment 6
Solution
1. Use integration by parts to solve:
R
(a) x ln xdx
R
(b) x2 e2x dx
Solution: a)
x2
2
ln x
1
2
; b)
e2x
2
x2
x+
1
2
2. Find the general solutions to the following ODEs:
(a) x0 (t)t = x (t) (1
t)
CAS EC 505 Mathematics for Economics
Assignment 4
Solution
1. Solve the following optimization program:
max [ln x + ln y] subject to x2 + y 2 = 1
x;y
Solution: (x ; y ) =
p
p
2
; 22
2
2. Solve:
max [x + y] subject to x2 + y = 1
x;y
Also nd the equation of
CAS EC 505 Mathematics for Economics
Assignment 5
Solution
1. Suppose f : R2 ! R is dened by:
x4 + a(x2
f (x; a) =
1)
where x 2 X = [ 1; 1] and where a 2 R:
(a) Find the solution correspondence x(a) for all a 2 R: Draw a graph.
(b) Find and draw a graph o