Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 1: Solutions
Due: September 15, 2010
1. (a) A B C
(b) (A B c C c ) (Ac B C c ) (Ac B c C ) (A
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
6.041/6.431 Fall 2010 Quiz 2 Solutions
Problem 1. (80 p oints) In this problem:
(i) X is a (continuous) u
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2009)
Quiz 2 Solutions:
November 3, 2009
Problem 2. (49 points)
(a) (7 points)
We start by recognizing that fX
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
6.041/6.431 Spring 2009 Quiz 1
Wednesday, March 11, 7:30  9:30 PM.
SOLUTIONS
Question
0
1
2
Name:
3
Recitation Instructor:
Part
Score
all
a
b
c
d
e
f
a
b
c
d
e
f
g
h
Total
Out of
2
40
5
2
5
5
5
5
2
2
2
5
5
5
5
5
100
Write your solutions in this quiz pa
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Quiz 1 Solutions:
October 12, 2010
Problem 1.
1. (10 p oints) Let Ri b e the amount of time Stephen spen
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2009)
Quiz 1 Solutions:
October 13, 2009
1. (10 p oints) We start rst by listing the following probabilities:
P
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
6.041/6.431 Spring 2009 Final Exam
Thursday, May 21, 1:30  4:30 PM.
Question
0
1
2
3
4
Name:
Recitation Instructor:
5
6
7
Part
Score
all
all
a
b
c
a
b
c
a
b
a
b
c
d
e
a
b
Total
Write your solutions in this quiz packet, only solutions in the quiz packet
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
6.041/6.431 Fall 2010 Final Exam Solutions
Wednesday, December 15, 9:00AM  12:00noon.
Problem 1. (32 p o
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2009)
Final Solutions:
December 15, 2009
Problem 2. (20 points)
(a) (5 points)
Were given that the joint PDF is
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 6: Solutions
1. Let us draw the region where fX,Y (x, y ) is nonzero:
y
2
1
yx=z
0
1
2
x
x=
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 5: Solutions
1. (a) Because of the required normalization property of any joint PDF,
2 2
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 4: Solutions
1. (a) From the joint PMF, there are six (x, y ) coordinate pairs with nonzero p
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 3 Solutions
Due September 29, 2010
1. The hats of n persons are thrown into a box. The person
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 2: Solutions
Due September 22, 2010
1. (a) The tree representation during the winter can be d
Probabilistic Systems Analysis and Applied Probability
EECS 6.431

Fall 2010
6.041/6.431 Spring 2008 Quiz 2
Wednesday, April 16, 7:30  9:30 PM.
SOLUTIONS
Name:
Question
0
1
2
Recitation Instructor:
TA:
3
6.041/6.431:
Part
Score
all
a
b
c
d
e
f
a
b
c
d
e
Total
Out of
3
36
4
5
5
8
5
6
4
6
6
6
6
100
Write your solutions in this qu