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Probability and Stochastic Processes 3rd Edition Solutions

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Probability and Stochastic Processes Solutions
A Friendly Introduction for Electrical and Computer Engineers
Third Edition
Roy D. Yates, David J. Goodman, David Famolari

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Problem Solutions – Chapter 1

Based on the Venn diagram on the right, the complete Gerlandas
pizza menu is
• Regular without toppings
• Regular with mushrooms
• Regular with onions
• Regular with mushrooms and onions
• 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
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. If this is clear, the questions are easy.
(a) Since N = T
, 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 Gerlanda’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)

Problem 1.1.3 Solution
At Ricardo’s, the pizza crust is either Roman (R) or Neapolitan
(N). To draw the Venn diagram on the right, we make the following observations:
• The set {R, N} is a partition so we can draw the Venn diagram with this
• 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




Probability and Stochastic Processes 3rd Edition Solutions

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