Probability and Statistics - The Science of Uncertainty
W. H. Freeman & Co.
This page contains links to various material relevant to
this text. If you have errata or comments please send these
to the authors.
Selected Answers
Selected answers to some of the problems in the text can be obtained
by downloading a pdf file here.
Errata and Minor Changes
All errata have been corrected in the second printing except those
preceded by (*)
- Chapter 1
- Challenge 1.5.15 - just before (a) change "roll" to "rolls".
- (*) Problem 1.64. - the sample space $S$ should contain the interval [0,1].
(thanks to Professor Chris Andrews)
- Chapter 2
- (*) Examples 2.3.4 and 2.3.5 - the illustrations in these examples that
refer to baseball should say that the number of at-bats without a hit
before the first hit follows a geometric distribution (Example 2.3.4),
and the number
of at-bats without a hit before the r-th hit follows
a negative-binomial distribution (Example 2.3.5). (Professor Chris Andrews)
- (*) Example 2.3.7 - The last sentence of the second paragraph
should read "This kind of sampling is called sampling without replacement."
(thanks to Sam Hong Li of York U.)
- (*) Challenge 2.4.17, p. 60, the change of variable should be u=x+y,
v=x/u. (thanks to Doug Rivers).
- Section 2.5, page 61, line 4 $B=(\infty,x]$ should be $B=(-\infty,x]$
(thanks to Eric Beiers).
- Section 2.5, page 65, in Example 2.5.2 towards the end, $\Phi(-0.96)=0.1469$ should be $\Phi(-0.96)=.1685$. This changes the final answer to
$\Phi(-0.965)=0.16725$ and $P(-0.63 <= X <= 2.0) = 0.6368 - 0.16725 = 0.46955$.
- (*) Exercise 2.5.2 "flipping" should be "rolling". (Thanks to Chris Andrews)
- (*) Problem 2.5.10, "Definition 2.5.4 should be "Definition 2.5.2".
(Thanks to Ping Gao)
- (*) Problem 2.6.8 - Should have $f_X(x+\mu)=f_X(\mu-x)$. (Thanks to Wenjie Qiao)
- (*) p. 93, lines 5 and 6 - These should read "conditioning on X=0.8 increses the probability that 0.2 <= Y <= 0.3, from about 0.0336 to 0.0398."
(Thanks to Wenjie Qiao)
- (*) p.100, line 5 "for 0<=n<=1" should be "for 0<=x<=1". (Thanks to Ping Gao)
- (*) Exercise 2.8.21 $X_{(i)} \leq i$ should be $X_{(i)} \leq x$. (Thanks to Chris Andrews)
- (*) p.106, line 1 "{(x,y):f(x,0>0)}" should be "{(x,y):f(x,y)>0}". (Thanks to Ping Gao)
- Example 2.9.2 - the expression for Y in terms of W and Z should read
"Y=\sqrt{(z-w)/2}". Also the expression given for $f_Z,W(z,w)$ at the bottom
of page 106 should have been divided by the Jacobian. As this leads to a number of changes we
reproduce this example here. (thanks to Michelle Alexopoulos and Xudong Li)
- Example 2.9.3 - the expression recorded for del h_1/del u_1 should
have the expression (-1/u_1^2) in it rather than (-u_1^2). (thanks to Leo Trottier)
- (*) p.113, line 8 both "p" should "\theta". (Thanks to Ping Gao)
- (*) Example 2.10.2 $U_1 > p$ should be $U_1 > \theta$. (Thanks to Chris Andrews)
- (*) p.121, second last line "{(x,y):f(x,0>0)}" should be "{(x,y):f(x,y)>0}".
- Chapter 3
- Example 3.1.4 E(W) = -5.2 (not 5.2) (thanks to Erich Beiers)
- (*) p. 139 - the sentence just before Example 3.2.7 should be just after Example 3.2.7.
- Problem 3.3.19 - should be "Multinomial(n,theta_1,theta_2,theta_3)">
- (*) Example 3.3.7 - in the middle of the page should have
(Thanks to Sam Hong Li)
- (*) The first part of problem 3.4.14 should read $R_Y(t)=t^a R_X(t^b)$.
(Thanks to Michael Andrushchenko.)
- (*) Example 3.5.9 - limits on the integral for Var(X | Y=y) should
be from 0 to y. (Thanks to Wenjie Qiao and Xudong Li)
- Chapter 4
- (*) The headers on the even numbered pages of Section 4.4 are
labelled 4.5. The headers on the even numbered pages of section
4.5 are labelled 4.4. (thanks to Chris Andrews)
- (*) p. 201 - both probability statements have $X_n,k$ and this should
be $X_k,n$. (thanks to Chris Andrews)
- (*) p. 202 - $X_n,r$ should be $X_r,n$. (thanks to Chris Andrews)
- (*) p. 214, l.9 - extra right bracket on $T_i = cos(U_i^2)sin(U_i^4)$.
- (*) p. 222, l. 7-8 - should indicate that X_1 and X_2 are independent
and $cX_1 + d ~ N(c mu_1 + d, c^2 sigma_1 ^2)$. (thanks to Chris Andrews)
- (*) p. 223 - two lines above Theorem 4.6.3 should read "(also see Table D.3 in Appendix D)". (Thanks to Sam Hong Li)
- (*) On p. 223, in the statement of Theorem 4.6.4, the exponent of (1/2)
should be 1/2, instead of -1/2. (thanks to Doug Rivers)
- p. 224 - first two integrals on this page should have lower limits
- infinity rather than infinity. (thanks to Leo Trottier)
- (*) p. 229, l. 17-18 should read "If $X ~ F(1,n)$ then $X^2 ~ t(n)$."
(thanks to Chris Andrews)
- (*) p. 235 middle of page; second sentence after equation 4.7.1:
"Furthermore, by Theorem 4.6.2, the expressions within brackets in
(4.6.2) are all ..." should be "Furthermore, by Theorem 4.6.2, the
expressions within brackets in (4.7.1) are all ..." (thanks to Chris Andrews)
- Chapter 5
- (*) p. 257 - the display in the middle of the page has the wrong
inequalities (thanks to Lizhen Xu). It should appear as shown below.
- Problems 5.5.10 and 5.5.11 - sigma_0 should be sigma.
- (*) p. 280: The problem after 5.5.18 should be 5.5.19 (not 5.4.19).
(thanks to Chris Andrews)
- Chapter 6
- (*) Example 6.1.3 - Likelihood should be denoted$L(\theta | 6 )$ since the data is $s=6$.
- (*) Example 6.1.3 - should have $L(\theta | 4 )= \binom{9}{6} ...$. (thanks
to Houssam G. Nassif)
- (*) Problem 6.1.17 - This should read ''Multinomial($1, \theta, 2\theta, 1 -
3\theta)$'' and the range of $\theta$ is [0,1/3]. (thanks to Chris Andrews)
- (*) Example 6.2.3 - First sentence should read "lifetime is known to be
distributed Exponential(1/\theta)".
- (*) p. 287 - The first line of the Proof of Theorem 6.1.1 should be
''when $T(s_1)=T(s_2)$'', not $s_1$ both times. (thanks to Chris Andrews)
- (*) p. 287 - Example 6.1.8, restriction on sigma should be ''$\sigma^2>0$''
not ''$\sigma\geq 0$''. (thanks to Chris Andrews)
- (*) p. 297: Caption of Figure 6.2.2 should have x_{(10)} = 1.3 (not 3.3).
(thanks to Chris Andrews)
- (*) p. 299 - on the 4th line should have $Var_\theta (X_i)$ not
$Var_\theta (X_i^2)$. (thanks to Houssam G. Nassif)
- (*) Problem 6.2.14 should be restricted to the discrete case.
- (*) Labels 6.4 and 6.5 are permuted in the running heads of these sections.
- (*) p. 312 - Equation at the bottom should be as follows. (thanks to Chris Andrews)
- p. 323, line 23 (6.3.13) "0.8" should be "0.99". (Thanks to Ping Gao)
- (*) p. 330, line 16, the equation in the middle of the page
$E_\theta ( X_1^{2i} - 2\mu_i^i X_1 + \mu_i^2 )$ should be
$E_\theta ( X_1^{2i} - 2\mu_i X_1^i + \mu_i^2 )$. (Thanks to Houssan G. Nassif)
- (*) p. 331, Example 6.4.1 - at the end of the first paragraph it should be
$1/\mu^2$ not $1/\mu$. (thanks to Chris Andrews)
- (*) p. 346, Example 6.5.3 - displayed equation is missing
$z_{(1+\gamma)/2}$ on the right.
(thanks to Chris Andrews)
- Chapter 7
- (*) p. 359, Exercise 7.1.3 - should read "what is the posterior probability
that $\mu$ is positive ..., $\mu_0 = 0$''. (thanks to Chris Andrews)
- (*) p. 360, Exercise 7.1.8 - should read "estimate the posterior probability
that the coefficient of variation is greater than .125''. (thanks to Chris Andrews)
- p.361, line 2. ".>0.125" should be ".>2" (the solutions are provided for the latter case) (Thanks to Ping Gao)
.
- (*) p. 374, Example 7.2.12 - in the final sentence the reference to
Example 7.2.14 should be to Example 7.2.13. (thanks to Chris Andrews)
- (*) p. 390, Example 7.3.2 - there is a missing parentheses as third
sentence should read "If $\lambda>2$ then ...
$\sigma(\lambda/(\lambda-2))^{1/2}$ ... ". (thanks to Chris Andrews)
- Chapter 8
- (*) p. 417, Problem 8.1.17 "N(\mu,\sigma^2)" should be "N(\mu,\sigma_0^2)" and "\theta" should be "\mu". (Thanks to Ping Gao)
- (*) p. 417, Problem 8.1.18 "N(\mu,\sigma^2)" should be "N(\mu,\sigma_0^2)".
(Thanks to Ping Gao)
- (*) p. 422, line -4 "\bar{X} ~N(\mu,\sigma_0^2)" should be "\bar{X} ~ N(\mu,\sigma_0^2/n)". (Thanks to Ping Gao)
- (*) p. 426, lines 1 and 2, should read "c'_0=3 because
P_.025 ( n\bar{X}>3)=1-0.962=0.038,
P_.025 ( n\bar{X}>2)=1-0.831=0.169, ...". Further the following lines should be
modified to reflect that we reject categorically when the number of successes is greater than 3, accept categorically when the number of successes is less
than 3 and randomly reject with probability 0.012 when the
number of successes equals 3. (Thanks to Ping Gao)
- (*) p. 428, Problem 8.2.4 "in (8.2.5)" should be "in (8.2.7)".
(Thanks to Ping Gao)
- (*) p. 433, Problem 8.3.9 "and and " should be "and". (Thanks to Ping
Gao)
- Chapter 9
- (*) p. 461, l. 14 - should read "x_1 + ... + x_k = n" rather than using
f_1 + ... + f_k =n.
- Chapter 10
- Example 10.1.1 - First sentence of second paragraph should read as follows.
"In this case, the conditional distributions have constant shape and change,
as $x$ changes, through the conditional mean."
- Example 10.1.2 - First sentence of second paragraph should read as follows.
"In this case, the conditional distributions have constant shape and change,
as $(x_1,...,x_k)$ changes, through the conditional mean,
which changes according to the
function $beta_1+beta_2 x_1+...+beta_{k+1} x_k$.
- (*) p. 494 - In the next to last table, \theta_{1a} should be \theta_{a1}
\theta_{b1} should be \theta_{1b}, and \theta_{ba} should be \theta_{ab}.
(thanks to Houssam G. Nassif)
- (*) Exercise 10.2.6 - Y should be called a response variable rather than a predictor.
- Section 10.3.2, page 507, first sentence of the third paragraph should
read as follows.
"In the regression model, we assume that the conditional
distributions have constant shape and change,
as we change $x$, through the conditional mean."
- (*) p. 511, l. 4 needs a final period.
- (*) Exercise 10.3.1 - should read Bernoulli($\theta$).
- Section 10.3.4, second sentence should read as follows.
"For the regression model, we assume that the conditional distributions of $Y$,
given the predictors, have constant shape and
change, as the predictors change, through the conditional mean
$E(Y|X_1=x_1,...,X_k=x_k)$."
- (*) p. 544 and p. 546 - two displayed formulas are missing the 1 in the
summation index, i.e, should read i=1 below the summation sign.
- Computer Exercise 10.5.4 - (d) second time should be (e).
- Chapter 11
- (*) Page 586, Problem 11.1.11: maximize P(\tau_c<\tau_o),
not P(\tau_0<\tau_c). (thanks to Professor Chris Andrews)
- (*) Page 590, Example 11.2.10, third line should be
P(X_0=1)=1/7, P(X_0=2)=2/7, P(X_0=3)=4/7 (and delete final right parenthesis).
(thanks to Chris Andrews)
- (*) Page 591, Example 11.2.11: in the second line from the
bottom of the page p_{11} p_{13} should be (0)(1/2) not (0)(1/4).
(thanks to Len Philipp)
- (*) Discussion Topic 11.2.16, "Example 12.2.9" should be "Example 11.2.9".
- (*) Page 609, Example 11.3.5, the last two lines of
display (11.3.5) should include the factor $(1+|y|)^3/(1+|x|)^3$ after the comma. (thanks to Nicholas Horton)
- (*) Page 612, Computer Problem 11.3.10 should refer to Exercise
11.3.5. (thanks to Chris Andrews)
- (*) Page 616, third line of section 11.4.3: E(X_T)=a (not 0).
(thanks to Chris Andrews)
- Appendices
- Page 641 - in the Taylor series definition about 2/3 of the way
down the page, "(x)" should be "(0)" (four times).
- (*) p. 644, Figure A.6.2 - The wrong region is shaded. The correct
integration region is the upper half of the rectangle [0,6]x[0,3]
bisected by the line through (0,0) and (6,3). (many thanks to Zeynep Kirkizoglu)
- (*) p. 653 and 654 - range of values for $\theta$ in geometric and negative binomial distributions should not include 0.
- (*) p. 654 - domain for Beta density should be x in [0,1].
- (*) p. 655 - F density should be (argument should be x and power on x is wrong) as shown below.