Probabilistic Systems Analysis and Applied Probability
EECS 6.041

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 6.041

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 6.041

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 6.041

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 6.041

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 6.041

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 6.041

Fall 2010
LECTURE 5
Random variables
Readings: Sections 2.12.3, start 2.4
An assignment of a value (number) to
every possible outcome
Lecture outline
Mathematically: A function
from the sample space to the real
numbers
Random variables
Probability mass functi
Probabilistic Systems Analysis and Applied Probability
EECS 6.041

Fall 2010
LECTURE 4
Discrete uniform law
Readings: Section 1.6
Let all sample points be equally likely
Then,
Lecture outline
P(A) =
Principles of counting
number of elements of A
A
=
total number of sample points

Just count. . .
Many examples
permutation
Probabilistic Systems Analysis and Applied Probability
EECS 6.041

Fall 2010
LECTURE 3
Models based on conditional
probabilities
Readings: Section 1.5
3 tosses of a biased coin:
P(H ) = p, P(T ) = 1 p
Review
Independence of two events
p
1 p
HHT
p
Independence of a collection of events
HHH
HTH
p
Review
p
1 p
1 p
P(A B )
P(A
Probabilistic Systems Analysis and Applied Probability
EECS 6.041

Fall 2010
LECTURE 2
Review of probability models
Readings: Sections 1.31.4
Sample space
Mutually exclusive
Collectively exhaustive
Lecture outline
Right granularity
Review
Event: Subset of the sample space
Conditional probability
Allocation of probabiliti
Probabilistic Systems Analysis and Applied Probability
EECS 6.041

Fall 2010
6.041 Probabilistic Systems Analysis
6.431 Applied Probability
Coursework
Quiz 1 (October 12, 12:0512:55pm)
Turn in recitation and tutorial scheduling form
(last sheet of course information handout)
40%
Weekly homework (best 9 of 10)
Pick up and
Probabilistic Systems Analysis and Applied Probability
EECS 6.041

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 6.041

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 6.041

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