Introduction to Computer Organization Fall 2013
Homework 1
Assigned: Thursday, Sept. 5, 2013
Deadline: Tuesday, Sept. 12, 2013
H ongji Wang
hongjiw
Name: _ Uniqname: _
Instructions:
1. Please write your name and uniqname in the spaces pro

EECS 301
Solutions to Homework 1
Assigned: January 13
Due: January 20 at 1:00pm electronically through gradescope.com.
1 Set operations and Venn diagrams
Express each of the following events in terms of the events A, B and C as well as the operations of
c

EECS 301 HW 2 Solutions
2.40 There are 8 possible numbers for the first digit, and 10 possible numbers for all the rest six
digits. Therefore, 8 106
2.44 310 = 59049 possible answers. P[a paper to have a certain answer sequence]= 3110 . Let X, Y
be answer

EECS 301 HW 3 Solutions
2.97
a)
P [0 or 1 error] = (1 p)100 + 100(1 p)99 p
= 0.3660 + 0.3697
= 0.7357
b) pk = P[Retransmission required) = 1 - P(0 or 1 error) = 0.2642
P(M retransmissions in total) = (1 pk )pm
k m = 1, 2,
P
P
j
j
m
m
P(M or more retransm

EECS 301 HW 4 Solutions
3.40
VAR[X] = E[X2] E[X]2
While E[X2] has g(X) = X2 the VAR[X] includes the term E[X]2. Since equation 3.31b applies to
the form E[g(X)] it cannot be directly applied to the VAR where E[X]2 is needed.
2
[]2 = (=1 [| ][ ]) 3.31
and

University of Michigan
WINTER 2017
EECS301: Solution to Homework 2
1. Elevator Problem[10 points each]
(a) Let
A=cfw_at least one person gets off on floor 15
B=cfw_at least one person gets off on floor 25
C=cfw_at least one person gets off on floor 35
E=c

University of Michigan
WINTER 2017
EECS301: Solution to Homework 3
1. (a) FALSE. The inclusion-exclusion principle says
P (A B|C) = P (A|C) + P (B|C) P (A B|C).
(b) TRUE. Note A C = cfw_b, c = B. We see that
1
20
= P (B) = P (A C). Moreover,
P (A)P (C) =

University of Michigan
WINTER 2017
EECS301: Solution to Homework 1
1 (a) A B C
(b) (Ac B c C c ) (A B c C c ) (Ac B C c ) (Ac B c C)
(c) (A B c C c ) (Ac B C c ) (Ac B c C)
(d) A (Ac B c )
(e) (A B C c )
The Venn diagrams are shown in Figures 1-5.
A
B
C
F

University of Michigan
WINTER 2017
EECS301: Solution to Homework 4
1 Let Ai be the event that component i works, and let B be the event that the system as a whole is
functioning.
(a)
P (B)
=
P (A1 A2 ) A3 )
=
P (A1 A2 ) + P (A3 ) P (A1 A2 A3 )
=
p2 + p p3

EECS 301 TOPICS INCLUDED IN THE MIDTERM
I.
Random Events
1. Fundamentals: The main axiom of probability theory; chance experiments; outcomes; sample
space; events; events classes; probability measure; Kolmogorovs axioms and their corollaries;
probability

EECS 301 HW 1 Solutions
Problem: Specify the sample space for each of the chance experiments
class.
introduced in
E1 - Toss a coin; S = cfw_H,T
E2 - Toss a 6 sided die; S = cfw_1, 2, 3, 4, 5, 6
E3 - Select a ball from an urn with the balls numbered 1-50;

University of Michigan
WINTER 2017
EECS301: Solution to Homework 10
1 Joint PDF
Note that the joint PDF takes value equal to 1 in the semi-circular region. This is follows from
Z 2 Z 2x2
1
dxdy = 1.
0
2x2
(a) Note that X takes values from 0 to 2. Hence

University of Michigan
WINTER 2017
EECS301: Homework 8
Assigned: March 24, 2017
Due: March 31, 2017 at 5PM in the 301 Box in EECS 2420
Text: Probability, statistics and random processes for electrical engineering by Alberto LeonGarcia (third edition)
Read

EECS 301 W17
April 07, 2017
Discussion Set 11
Solutions
Abhinav Sinha (GSI)
1. For X , Y independent random variables, show that Var(X + Y ) = Var(X ) + Var(Y ).
Sol.
Var(X + Y ) = E (X + Y )2 E [X + Y ]2
2
= E X 2 + 2X Y + Y 2 E [X ] + E [Y ]
(1a)
(1b)
=

EECS 301 W17
March 31, 2017
Discussion Set 10
Solutions
Abhinav Sinha (GSI)
An application of the Central Limit Theorem
1. 52% of the residents of New York City are in favor of outlawing cigarette smoking on
university campuses. Approximate the probabilit

University of Michigan
WINTER 2017
EECS301: Solutions to Practice Problems for Final Exam
1 Joint PMF
The joint PMF of X and Y is given in the following table, where the columns correspond to
different values of Y and rows correspond to different values o

EECS 301 W17
March 24, 2017
Discussion Set 9
Solutions
Abhinav Sinha (GSI)
A useful formula for calculating moments
1. Suppose X; Y are non-negative random variables, show part a) below and then use it to
prove part b).
a)
b)
E[
Xn
P (Y > y) dy:
(1)
nxn 1

University of Michigan
WINTER 2017
EECS301: Solution to Homework 8
1. Median
The median is the value of x such that FX (x) = 1/2.
(a) FX (x) = 1 ex = 1/2 x = (ln 2)/.
(b) Note that the PDF of a Gaussian random variable with mean is given by
fX (x) =
1
2

University of Michigan
WINTER 2017
EECS301: Homework 9
Assigned: March 31, 2017
Due: April 7, 2017 at 5PM in the EECS301 Box in EECS 2420
Text: Probability, statistics and random processes for electrical engineering by Alberto LeonGarcia (third edition)
R

EECS 301 W17
March 31, 2017
Discussion Set 10
Abhinav Sinha (GSI)
An application of the Central Limit Theorem
1. 52% of the residents of New York City are in favor of outlawing cigarette smoking on
university campuses. Approximate the probability that mor

University of Michigan
WINTER 2017
EECS301: Solution to Homework 9
1. IC testing
For each test, there are three outcomes of interest: A = a faulty IC-1 is detected, B = a faulty IC-2 is
detected, and C = no faulty IC of either kind is detected. Imagine fi

University of Michigan
WINTER 2017
EECS301: Practice Problems for Final Exam
This is a collection of old exam problems, old homework problems, and other problems that I
think will help you understand the material and prepare for the exam. I also strongly

EECS 301 W17
March 24, 2017
Discussion Set 9
Abhinav Sinha (GSI)
A useful formula for calculating moments
1. Suppose X; Y are non-negative random variables, show part a) below and then use it to
prove part b).
a)
E[Y ] =
b)
E[X n ] =
Z1
P (Y > y) dy:
(1)

EECS 301 W17
April 07, 2017
Discussion Set 11
Abhinav Sinha (GSI)
1. For X , Y independent random variables, show that Var(X + Y ) = Var(X ) + Var(Y ).
2. N is a discrete uniform random variable that takes values in the set
Zq = cfw_0, 1, 2, . . . , q 1,

University of Michigan
WINTER 2017
EECS301: Homework 10
Assigned: April 7, 2017
Due: April 14, 2017 at 5PM in the 301 Box in EECS 2420
Text: Probability, statistics and random processes for electrical engineering by Alberto LeonGarcia (third edition)
Read

ECE-340
Spring 2008
Probabilistic Methods in Engineering (3 credits)
M, W 3:00-4:15 PM
Room: Dane Smith Hall 325
Syllabus
Course Goals: To introduce the student to basic theoretical concepts and computational tools in
probability and statistics with empha