# Errata for 'Mathematical Foundations of Computer Networking'

My apologies for the following errors in the book.

## Chapter 1: Probability

• Example 1.15, Page 12: The fact “P(52) = 0.54” is not known at this point in the book. It is derived in Example 1.16.
• Section 1.3.4, Eq. 1.19, Page 20: E[g(C)] should read E[g(XC)]
• Section 1.3.5, The equation for variance should read V[X] = (Σ xi2)/n - (Σ (xi/n))2. For practical use of this formula, please refer to this site.
• Section 1.4.1 Note that skewness is defined as M3μ/(M2μ)1/2
• Section 1.5.1, Page 25, last sentence: Variance should be a(1-a)
• Section 1.6.3 Example 1.33, the calculation should read 1-F(6) = 1- 1- e-6/3 = e-2 = 13.5%
• Section 1.7.3 Example 1.38 Page 39, the last equation should read 1-(p(X=0) + p(X=1)... rather than (1-p(X=0) + p(X=1) ...
• Section 1.8.1, Page 47, the LHS of the last equation on the page should read p(L|R) not p(PLR)
• Exercise 13 should read "Prove ...N(0,1) without using MGFs.

## Chapter 2: Statistics

• Section 2.4.5 Example 2.10 has an error in computation. s2 should be 0.243, s should be 0.493, so the t variate is 0.016 (not 0.0048).
• Section 2.4.5 Equation 2.23 has an error: in the denominator, the left hand term's denominator should read n(n-1) and the right hand term's denominator should read m(m-1)
• Section 2.4.6 In equation 2.24, the LHS should read P(o1n1o2n2...oknk)
• Section 2.4.6, Example 2.12: "...any value greater than 3.84 occurs with probability less than 95%" - this should read 5%
• Section 2.5 Example 2.17: The mean throughput should read 60.4kbps and mean delay 766.66 ms. The value of r should be -0.82.
• Exercise 5: this exercise is incorrect, since the Chi-squared test should be done using counts, not percentages.

## Chapter 6: Queueing theory

• Section 6.1.2, Page 238-239: The terms “waiting time” can mean different things. In the theorem as well as this proof, it refers to the time waiting in the queue, but not for service. However, the theorem applies to the latter case as well.
• Section 6.2 On pages 240/241, the subscript for X should be k rather than i, to avoid confusion later, where i is used to refer to the state, rather than time step
• Section 6.2.1 Example 6.5: The example should read 'Continuous Space Process'
• Section 6.2.3, Equation 6.2, Page 246: The suffix of the first p in the sum should read “ik” instead of “ij”.
• Section 6.2.8, Theorem 6.11 is actually Theorem 6.1
• Section 6.2.11 ...the probability of staying in the state for m steps is given by pjjm-1(1-pjj) not pjjm(1-pjj)
• Section 6.3.3, Page 254, first sentence after Equation 6.9: Should read “during a small interval” instead of “during any interval”.
• Section 6.3.3, Page 254, first sentence after Equation 6.9: Should read “conditional on its already being at state i at time t” instead of “conditional on its already being at state i”.
• Equation 6.11, Page 254 “(t)” should be removed from the equation
• Section 6.4, Page 255, last sentence before Equation 6.15: Should read “Given this fact, Equation 6.10, and Equation 6.14, we find that”
• Section 6.4.5, Page 261, just after Eq. 6.30, the condition for convergence is the existence of a value j0 such that for all values of j > j0 lambdaj < muj+1
• Example 6.22, Page 264, first sentence: Reference should go to Example 6.21, not “6.19”.
• Section 6.5, Page 265, just before Eq. 6.38, the equation is (rho/(1-rho)) not (rho/(1-rho)c
• Example 6.23, Page 266, first sentence: Reference should go to Example 6.21, not “6.20”.
• Example 6.25, Page 269, first sentence: Reference should go to Example 6.21, not “Equation 6.21”.

## Chapter 7: Game theory

• Section 7.2, Page 292, 4th line from bottom should read "If either prisoner informs on the other (defects, or D), that prisoner is set free, ...
• Section 7.2, Page 293, payoff matrix for Prisoner's Dilemma game, the tuple in row 1 col 2 should read (-4, 0)
• Section 7.2.4, Page 294, just before example 7.18, for the maximin strategy, the equation should read

$\pi(s^{min}_{-i}(s_i^*), s_i^*) /geq \pi(s^{min}_{-i}(s_i^j), s_i^j) \forall s_i^j \neq s_i^*$

• Section 7.3.1, Page 302: an English auction does not reveal the winner's true valuation, since this valuation can be higher than the winning bid
• Section 7.3.2, Page 303: In the discussion on independence of irrelevant alternatives, the text should read "...the other alternatives could be arbitrarily ranked, but as long as a majority of individuals rank a higher than b, so too should the social-welfare function."
• Section 7.3.3, Page 306: The utility of a bid is given by v-p, not v-b
• Section 7.3.7, Page 310, Eq. 7.6: there are extra parentheses that can be removed
• Solution to Exercise 5, Page 445: this solution is incorrect. Instead, there should be 8 rows, corresponding to the pure strategies (Y,Y,Y), (Y,Y,N), (Y,N,Y), ... (N,N,N). For each cell, however, the payoffs are as shown in the table on page 445. For example, for the row (Y,Y,Y), for column L,the payoffs are (1, a-1), for column M, the payoffs are (2, a-2), and for column H, the payoffs are (3, a-3). This reflects the idea that the row player can choose one of their 8 strategies at the start of the game and have it played for them no matter what the column player does.
• Solution to Exercise 12, Page 446: should read "Any mixed strategy of R2 and R3 that plays R2 with a probability ..." instead of plays R3.
• Exercise 18, Page 318 has some extra '*' symbols that should be ignored. The solution on page 447 also has an error. v2(x-1) + v3(x-1) -(v2(x*) +v3(x-*)) = 120(1-e-0.5*2.197) - 40*2.197 -( 120(1-e-0.5*2.5055) - 40*2.5055) = -17.838. Similarly, the corrected values of p2 and p3 are -23.559 and -34.977.

My thanks to the following readers for taking the time to point out these errors:

• Martin Jacobsson, Communication Research (CoRe), Dept. of IT, Uppsala University, Sweden
• Jinwen Lu, Yerbol Aussat, Qiushi Jiang, Anthony Anthony University of Waterloo