# Probability and Statistics - The Science of Uncertainty, Second Edition

## by Michael J. Evans and Jeffrey S. Rosenthal We are pleased to now make the book available for free. If you are an instructor and would like a copy of the solutions manual please email one of the authors. The book is available as a single pdf file or by individual chapters. Also, a version with a clickable TOC to help navigate through the book is available here (with thanks to Callum Cassidy-Nolan).

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.

If you have errata or comments please send these to the authors.

## Errata and Minor Changes

• Chapter 1

• p.7, last display - should read A \cap B^c = \{s : s\in A \hbox{ and } s \n otin B \}
• p.7, last display - should read A \cap B^c = \{s : s\in A \hbox{ and } s \notin B \}

• p. 21, in the proof of Theorem 1.5.1 the formula should be P(A_i\cap B)=P(A_i)P(B|A_i) (thanks to Tom Wehrly)

• Chapter 2

• p. 34 - Example 2.1.2, should say Y(clear) = 7/8 instead of Y(rain)=7/8. (thanks to Ryan Dubay)

• p. 39 - it seems that the Kindle version is missing this page. (thanks to Steven Kern)

• p. 73, Exercise 2.5.8, - the definition of F_Y should be changed to F_Y(y)=1-(1-y)^3 for 1/2 \leq y \leq 1. (thanks to Thomas Wehrly and his students)

• p. 83 Figure 2.7.1 is actually the plot of $F_{X,Y} (x,y)= x^2y$. (thanks to Nick Souza)

• p. 124 Exercise 2.10.8 the density should be 3sqrt(x)/2 (thanks to Jason Hrncir)

• Chapter 3

• Some of the Theorems concerning expectations do not state that the relevant expectations actually exist and this should be considered as being implicit, e.g. Theorems 3.1.2, 3.1.3, 3.1.4, 3.2.2, 3.2.3, 3.2.4. (thanks to Csaba Szepesvari)

• p. 149 in display 3.3.1, \sigma^2_x should be \sigma^2_X. (thanks to Thomas Wehrly and his students)

• p. 150 In the second last line on this page the ;ast equality should be E((aX+b-a\mu_X-b)^2) as there is a minus sign missing on the last b. (thanks to Jared Becksfort)

• p.156 Definition 3.3.4 should have the condition that both variances are positive since Corr is not defined when either variance equals zero. (thanks to Thomas Wehrly and his students)

• p. 166, Example 3.4.8 The equation Var(X) = E(X^2) - (E(X)) should have (E(X))^2 instead of (E(X)). (thanks to Ray Hinton)

• Chapter 4

• p.216, the sentence under Corollary 4.4.3, \sqrt{n}(M_n-n\mu) should be \sqrt{n}(M_n-\mu) (thanks to Masashi Yoshioka, UCLA)

• p. 223, Exercise 4.4.14 - \theta should be \lambda. (thanks to Thomas Wehrly and his students)

• p. 243, Exercise 4.6.3. For a unique solution assume C_1 \ne 0 and C_3 \ne 0. (thanks to Thomas Wehrly and his students)

• p. 243, Exercise 4.6.8. Displayed formula should be C1(X+C2)^C3/((Y+C4)^2)^C5 ~ t(C6) (note that Y+C4 in the denominator is now squared).

• p. 251, there are a number of typos so we have reproduced the last half of the page here. (thanks to Thomas Wehrly and his students)

• Chapter 5

• Example 5.3.2 - the Exponential(1.5) distribution should have been labelled as the Exponential(2/3) distribution to reflect the fact that we parameterize the exponential distribution in the book by the rate rather than the mean. (thanks to Daren Cline)

• p. 280 in problem 5.4.15(d). The factor (N-n)/(N-1) is referred to as the "finite sample correction factor" and should have been called the "finite population correction factor". (thanks to Carl Mueller)

• Chapter 6

• p. 318, problem 6.2.10 should refer back to problem 2.6.17 not 2.6.12. (thanks to Noah Alderton)

• p. 355, formula for bootstrap percentile confidence interval should have left-hand endpoint equal to \hat{psi}_{(1-\gamma)/2}. (thanks to Thomas Wehrly and his students)

• Chapter 7

• p. 380, l. 4 The comment in parentheses should read "recall that \$\theta_k= 1 - \theta_1 ... - \theta_{k-1}". (thanks to Raymond Hinton).

• p. 385 The reference to Problem 4.6.13 at the bottom of this page (3 lines from the bottom) should be to Problem 4.6.16. (Thanks to Zach Fuller)

• p. 424 - the fourth line should read nx_{bar} = 5, instead of nx_{bar} = 2.3 (thanks to Travis Lilley)

• Chapter 8

• p. 458, Problem 8.2.16 - First part of the problem should have (\sigma_0)^2 < (\sigma_1)^2. (thanks to Daren Cline and his students)

• Chapter 9

• Chapter 10

• Chapter 11

• p. 629, in Example 11.2.12 P_1(X_3=2) = \sum_{k \in S} \sum_{l \in S} p_{1k} p_{kl} p_{l3} should be P_1(X_3=2) = \sum_{k \in S} \sum_{l \in S} p_{1k} p_{kl} p_{l2}. (thanks to Kent Gauen)

• Appendices

• p. 681 - in the last display on this page the limits of integration in the inner integral should be 2y to 6. (thanks to Jared Becksfort)

• p. 730, solution to 2.1.7 (a) should be W(1) = 2 and solution to (c) should be W(3) = -1. (thanks to Thomas Wehrly and his students)

• p. 731, solution to 2.4.3(d) should be e^{-4*(26)^{1/4}}. (thanks to Thomas Wehrly and his students and to Anthony Daspit)

• p. 731, solution to 2.5.1 should read "since F_X(x)=0 for x <1/6 and F_X(x)=1 for x>=1". (thanks to Jason Hrncir)

• p. 731, solution to 2.5.13(c) should be P(-1 < X <1/2) = 3/4, P(X = 2/5) = 5/12, P(X = 4/5) = 1/4.

• p. 731, solution to 2.5.15 (a) (2/3)e^{-16/25} (b) 11/12 - (2/3)exp{-1/2} (f) 5/36. (thanks to Jason Hrincir)

• p. 731, solution to 2.6.9(b) density should be f_Z (z) = z^7 /2 for 0 < z < \sqrt(2). (thanks to Thomas Wehrly and his students)

• p. 731, solution to 2.6.11 should read "for 0 < y < \pi^2 and 0 otherwise". (thanks to Jason Hrncir)

• p. 732, solution to 2.8.7(c) and (d) change 50,000 to 500,000 (thanks to Thomas Wehrly and Jason Hrncir)

• p. 732, solution to 2.9.1, in the partial derivatives for h_1 and h_2 wrt u_2 the factor of 2 multiplying log(1/u_1) should not be there. (thanks to Jason Hrncir)

• p. 733, solution to 3.1.3(a) should be E(X) = -1. (thanks to Thomas Wehrly and his students)

• p. 733, solution to 3.2.5 should be -77/3. (thanks to Daren Cline and his students)

• p. 734, solution to 3.3.11 should be E(X) = 7/2, E(Y) = 7, E(XY) = 329/12, Cov(X,Y) = 32/12. (thanks to Thomas Wehrly and his students)

• p. 734, solution to 3.3.13 should be Cov(Z,W) = 1/36, Corr(Z,W) = 1/17. (thanks to Thomas Wehrly and his students)

• p. 734, solution to 3.5.11 - the corrected solution can be found here. (thanks to Daren Cline and his students)

• p. 735, solution to 4.1.1 should have P(Y-3 = 1) = 1/8 and should include P(Y_3 = 2 ) = 1/64. (thanks to Thomas Wehrly and his students)

• p. 736, solution to exercise 4.6.1. The mean of U is 43, not 44. (thanks to Darren Cline and his students)

• p. 736, solution to 4.6.3 shoud have C_1 = 1/5 and C_3 = 1/2. (thanks to Thomas Wehrly and his students)

• p. 737, solution to 5.5.1(d) should have the mean as 1.667. (thanks to Thomas Wehrly and his students)

• p. 739, solution to Exercise 7.1.1. (thanks to Daren Cline)