, Probability and Stochastic Processes
A Friendly Introduction for Electrical and Computer Engineers
Third Edition
INSTRUCTOR’S SOLUTION MANUAL
Roy D. Yates, David J. Goodman, David Famolari
September 8, 2014
Comments on this Solutions Manual
• This solution manual is mostly complete. Please send error reports, suggestions, and
comments to .
• To make solution sets for your class, use the Solution Set Constructor at the instruc-
tors site www.winlab.rutgers.edu/probsolns.
• Send email to for access to the instructors site.
• Matlab functions written as solutions to homework problems can be found in the
archive matsoln3e.zip (available to instructors). Other Matlab functions used in
the text or in these homework solutions can be found in the archive matcode3e.zip.
The .m files in matcode3e are available for download from the Wiley website. Two
other documents of interest are also available for download:
– A manual probmatlab3e.pdf describing the matcode3e .m functions
– The quiz solutions manual quizsol.pdf.
• This manual uses a page size matched to the screen of an iPad tablet. If you do
print on paper and you have good eyesight, you may wish to print two pages per
sheet in landscape mode. On the other hand, a “Fit to Paper” printing option will
create “Large Print” output.
1
,Problem Solutions – Chapter 1
Problem 1.1.1 Solution
Based on the Venn diagram on the right, the complete Gerlandas
pizza menu is
• Regular without toppings M O
• Regular with mushrooms
• Regular with onions
• Regular with mushrooms and onions T
• Tuscan without toppings
• Tuscan with mushrooms
Problem 1.1.2 Solution
Based on the Venn diagram on the right, the answers are mostly
fairly straightforward. The only trickiness is that a pizza is either M O
Tuscan (T ) or Neapolitan (N ) so {N, T } is a partition but they
are not depicted as a partition. Specifically, the event N is the
region of the Venn diagram outside of the “square block” of event T
T . If this is clear, the questions are easy.
(a) Since N = T c , N ∩ M 6= φ. Thus N and M are not mutually exclusive.
(b) Every pizza is either Neapolitan (N ), or Tuscan (T ). Hence N ∪ T = S so
that N and T are collectively exhaustive. Thus its also (trivially) true that
N ∪ T ∪ M = S. That is, R, T and M are also collectively exhaustive.
(c) From the Venn diagram, T and O are mutually exclusive. In words, this
means that Tuscan pizzas never have onions or pizzas with onions are never
Tuscan. As an aside, “Tuscan” is a fake pizza designation; one shouldn’t
conclude that people from Tuscany actually dislike onions.
(d) From the Venn diagram, M ∩ T and O are mutually exclusive. Thus Ger-
landa’s doesn’t make Tuscan pizza with mushrooms and onions.
(e) Yes. In terms of the Venn diagram, these pizzas are in the set (T ∪ M ∪ O)c .
2
, Problem 1.1.3 Solution
R N
At Ricardo’s, the pizza crust is either Roman (R) or Neapolitan
M
(N ). To draw the Venn diagram on the right, we make the fol-
lowing observations: W O
• The set {R, N } is a partition so we can draw the Venn diagram with this
partition.
• Only Roman pizzas can be white. Hence W ⊂ R.
• Only a Neapolitan pizza can have onions. Hence O ⊂ N .
• Both Neapolitan and Roman pizzas can have mushrooms so that event M
straddles the {R, N } partition.
• The Neapolitan pizza can have both mushrooms and onions so M ∩ O cannot
be empty.
• The problem statement does not preclude putting mushrooms on a white
Roman pizza. Hence the intersection W ∩ M should not be empty.
Problem 1.2.1 Solution
(a) An outcome specifies whether the connection speed is high (h), medium (m),
or low (l) speed, and whether the signal is a mouse click (c) or a tweet (t).
The sample space is
S = {ht, hc, mt, mc, lt, lc} . (1)
(b) The event that the wi-fi connection is medium speed is A1 = {mt, mc}.
(c) The event that a signal is a mouse click is A2 = {hc, mc, lc}.
(d) The event that a connection is either high speed or low speed is A3 =
{ht, hc, lt, lc}.
3
A Friendly Introduction for Electrical and Computer Engineers
Third Edition
INSTRUCTOR’S SOLUTION MANUAL
Roy D. Yates, David J. Goodman, David Famolari
September 8, 2014
Comments on this Solutions Manual
• This solution manual is mostly complete. Please send error reports, suggestions, and
comments to .
• To make solution sets for your class, use the Solution Set Constructor at the instruc-
tors site www.winlab.rutgers.edu/probsolns.
• Send email to for access to the instructors site.
• Matlab functions written as solutions to homework problems can be found in the
archive matsoln3e.zip (available to instructors). Other Matlab functions used in
the text or in these homework solutions can be found in the archive matcode3e.zip.
The .m files in matcode3e are available for download from the Wiley website. Two
other documents of interest are also available for download:
– A manual probmatlab3e.pdf describing the matcode3e .m functions
– The quiz solutions manual quizsol.pdf.
• This manual uses a page size matched to the screen of an iPad tablet. If you do
print on paper and you have good eyesight, you may wish to print two pages per
sheet in landscape mode. On the other hand, a “Fit to Paper” printing option will
create “Large Print” output.
1
,Problem Solutions – Chapter 1
Problem 1.1.1 Solution
Based on the Venn diagram on the right, the complete Gerlandas
pizza menu is
• Regular without toppings M O
• Regular with mushrooms
• Regular with onions
• Regular with mushrooms and onions T
• Tuscan without toppings
• Tuscan with mushrooms
Problem 1.1.2 Solution
Based on the Venn diagram on the right, the answers are mostly
fairly straightforward. The only trickiness is that a pizza is either M O
Tuscan (T ) or Neapolitan (N ) so {N, T } is a partition but they
are not depicted as a partition. Specifically, the event N is the
region of the Venn diagram outside of the “square block” of event T
T . If this is clear, the questions are easy.
(a) Since N = T c , N ∩ M 6= φ. Thus N and M are not mutually exclusive.
(b) Every pizza is either Neapolitan (N ), or Tuscan (T ). Hence N ∪ T = S so
that N and T are collectively exhaustive. Thus its also (trivially) true that
N ∪ T ∪ M = S. That is, R, T and M are also collectively exhaustive.
(c) From the Venn diagram, T and O are mutually exclusive. In words, this
means that Tuscan pizzas never have onions or pizzas with onions are never
Tuscan. As an aside, “Tuscan” is a fake pizza designation; one shouldn’t
conclude that people from Tuscany actually dislike onions.
(d) From the Venn diagram, M ∩ T and O are mutually exclusive. Thus Ger-
landa’s doesn’t make Tuscan pizza with mushrooms and onions.
(e) Yes. In terms of the Venn diagram, these pizzas are in the set (T ∪ M ∪ O)c .
2
, Problem 1.1.3 Solution
R N
At Ricardo’s, the pizza crust is either Roman (R) or Neapolitan
M
(N ). To draw the Venn diagram on the right, we make the fol-
lowing observations: W O
• The set {R, N } is a partition so we can draw the Venn diagram with this
partition.
• Only Roman pizzas can be white. Hence W ⊂ R.
• Only a Neapolitan pizza can have onions. Hence O ⊂ N .
• Both Neapolitan and Roman pizzas can have mushrooms so that event M
straddles the {R, N } partition.
• The Neapolitan pizza can have both mushrooms and onions so M ∩ O cannot
be empty.
• The problem statement does not preclude putting mushrooms on a white
Roman pizza. Hence the intersection W ∩ M should not be empty.
Problem 1.2.1 Solution
(a) An outcome specifies whether the connection speed is high (h), medium (m),
or low (l) speed, and whether the signal is a mouse click (c) or a tweet (t).
The sample space is
S = {ht, hc, mt, mc, lt, lc} . (1)
(b) The event that the wi-fi connection is medium speed is A1 = {mt, mc}.
(c) The event that a signal is a mouse click is A2 = {hc, mc, lc}.
(d) The event that a connection is either high speed or low speed is A3 =
{ht, hc, lt, lc}.
3
