Probability with Applications and R: Errata Updated: October 27, 2016

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1 Probability with Applications and R: Errata Updated: October 7, 06 Page xii, lines -4: The first sentence should be changed as: The book references numerous R script files which are available for download. It also includes many short R one-liners that are easily shown..... Page 7, top line: change TTTT to TTT Page 8, in Figure., the caption of the bottom, center Venn diagram: change excllusive to exclusive Page 0, 3rd line under the nd display: change =.0 >. to =.5 >. Page, 3rd line: Change GGG. to GOG. Page 5, nd math display: Change i= Ac to i= Ac i Page 9, Example.3. Starting from the last two lines, change the rest of the example to: written as {Y > 4}. The complementary event is {Y 4}, with P (Y 4) = P (Y = or Y = 3 or Y = 4) = P (Y = ) + P (Y = 3) + P (Y = 4) = P ({(, )}) + P ({(, ), (, )}) + P ({(, 3), (, ), (3, )}) = = 6. The desired probability is P (Y > 4) = P (Y 4) = (/6) = 5/6. Page 8, exercise., nd line: change Show the to Show that. Page 3,.34: 4th line: change 6 to :6; 6th line: change 6 to :6; 9th line: change -9 to 9 to Page 33,.40. The display should be: P (A) + P (B) + P (C) P (AB) P (AC) P (BC) + 3P (ABC). Page 33, exercise.43. The exercise should read: Use R to simulate the probability of two rolled dice having values that sum to 8. Page 58, 7th paragraph. The 4th and 5th sentences should be: The 90% specificity rate means that P (S c D c ) = And the false-positive rate gives P (S D c ) = 0.0. Page 59, Example.6. The last display should be = (0.07)(0.49) =

2 Page 66, Exercise.4. This exercise should read: According to the National Cancer Institute, for women aged 50 there is a.38% risk (probability) of developing breast cancer. Screening mammography has a sensitivity of about 85% for women aged 50, and a 95% specificity. That is, the false-negative rate is 5% and the false-positive rate is 5%. If a woman aged 50 has a mammogram and it comes back positive for breast cancer, what is the probability that she has the disease? Page 7, example 3.4, 4th line: change receive plasma from anyone to donate plasma to anyone Page 8, nd paragraph, the first sentence should be: It follows that the number of lists of length k made up of the elements {,..., n} is equal to k! times the number of k-element subsets of {,..., n}. Page 89, Last line of the nd bullet on the center of the page. Change p = 0.00 to p = 0.0 Page 99, Example 3.9, 3rd line: Change United States Assuming to United States. Assuming Page 99, Example 3.9. In the 3rd line of the display, change the last term from e 33/43 (33/43) to e 33/43 (33/43) Page 00, In the R box, 4th line from the bottom: Change 0.06 to 0.6 Page 0. In the first display change ( 0 9 ) 0 9 k to ( 0 9) k Page 08, nd line of the Independent random variables bullet: Change P (X = i)p (X = j) to P (X = i)p (Y = j). page, top of page. Change exercise (b) to: If 0 shoes are picked, what is the chance that at most one shoe from each of the 40 pairs will be picked? (Remember, a left shoe is different than a right shoe.) Page, exercise 3.6: In the nd sentence of the prompt change arranged in decreasing order to arranged in increasing order Page 3, Exercise 3.3(b): Change is a 60% she to is a 60% chance she Page 3, Exercise 3.6, 5th line: Change present at a particular site. to present in a region. Page 4, bottom line: change to 6) for k =,..., 6. Page, last line of the bottom display. After the first equality change P (X = ) to P (X = 4) Page 6, bottom line of the page: The righthand side of the last equality should be 9/36, not 5/6.

3 Page 35, Example 4.8: The first line of the first display should be k k P (X + Y = k) = P (X = i)p (Y = k i) = i= At the bottom of the example, the last display should be P (X + Y = k) = i= ( n ) ( ) k n n. { (k )/n, for k =,..., n (n k + )/n, for k = n +,..., n. Page 46, the line above equation (4.): Change (E[Y µ Y ) to (E[Y ] µ Y ) Page 47, nd line under the first display: change doubts about V ]X Y ], to doubts about V [X Y ], Page 53, Example 4.3. The last two lines should be = = 800, with standard deviation SD[Z] = 800 = Page 6. In the first R code block change from yol <- sample(:x,) yol <- sample(:xav,) Page 63, 3rd display, after nd equality should be to σ V [X µ] = σ σ =. Page 65, 3rd line of the display should be = y yp ( X f (y) ) Page 74, Exercise Change the first sentence from Simulate the dice game in Example 4.3. to Simulate the dice game in Exercise 4.3. Page 80, st paragraph of Section 5.., st line: change of tigers n to of tigers t. Also change the 3rd line of that paragraph to read: number of different tigers observed was d. From t and d the total number of tigers in Page 93. First lines of the variance calculation in the display above Example 5.: Change the minus sign before (n n) to a plus sign. Page 09, 4th line under the heading Other: Change n = 30 to n = 4 Page 0, Exercise 5.38, 4th line: change both references from 5 to 0 Page 0, Exercise 5.39, st line: Change in Example 5.. to in Example 5.6. Page 9, st display. At the end of the display equation, change = 0.85 to =

4 Page 3, Example 6.0, first display: The last expression on the right side of the display should be c not 3 3 Page 37, bottom math display should be: P (X < Y ) = (x,y):x<y Page 38, top math display should be: P (X < Y ) = / f(x, y) dx dy = y f(x, y) dx dy. s / s dt ds + s π / s dt ds. π Page 40, Example 6., nd display, st line: The dy dx term in the middle integral expression should be dx dy Page 60, add the following to the end of the nd paragraph: A probability model is obtained for a random position of the needle by letting D Unif(0, /) and Θ Unif(0, π), with D and Θ independent. Page 64. The last line should be = Page 69, Exercise 6.8. The density function should be f(x) = (x E[X])(y E[Y ])f(x, y) dx dy. e (e ) x e x /, for < x <. Page 7, Exercise 6.6. The density function should be f(x) = 4/(3x ), for < x < 4, and 0, otherwise. Page 80, left hand of top display, change to P (90 X 0) Page 89, nd math display, bottom integral should be = t Page 34. The bottom display should be t t v t v πσ e (h +v )/σ dh dv. E[S 30 S 7 = 400] = (30 7)µ = (50) = $550. Page 343, Example 8.8, nd line of the display. The constant in front of the integral should be λ F not λ Z Page 349, top math display, nd expression should be E[E[Z N]] (not E[E[Z T ]]) 4

5 Page 356, Exercise 8., first line: Change Y Unif(x) to Y Unif(0, x) Page 357, 8.8(d): change ln 4/9 to (ln 4)/9. Page 388, Example 9.9. The integral limits in the first two lines of the display should be 0 not. Also, the last line of the page should read: defined for all t < λ. Page 397, Exercise 9.33: The middle expression in Equation 9.5 should be E[e tx ] e tx Also, change the next line to assuming that E[e tx ] exists, where m is the mgf of X. Page 450: At the beginning of the nd display of R code, add > dog <- seq(,5,) Page 455: In section 9, the first comment should read # tabulates the number of each distinct entry in a vector Page 456: In the nd and 3rd displays, there is no curly brace at the end of the line of R code. Page 457: In exercise A.5, the nd line of the R command display should be prob=c(0.49,0.5)) Page 466, top line: Change of Y in and to of Y, and Page 470,.9. The solution is incorrect. Change /(n n n k + ) to /k!. Page 470, Combine.(a) and. (b) on the same line. Should be. (a) 0.90; (b) 0. Page 470,,7(ii): 5/8 (not 3/8). Page 470,.9(a), the set at the end of the last display: change {pq, nq, dq, qq, qd, qn, qnqp} to {pq, nq, dq, qq, qd, qn, qp}. Page 470,.35: 3/60. Page 47, Solution 3.5 should be Page 47, Exercise 3.5: (a) should be Page 47, Exercise 3.5: Change (c) 0.365; (d) to (c) and Page 473, Exercise 4.(d) solution should be , not 0.95 Page 473, Exercise 4.3(b) solution should be 0.5, not Page 474, Solution to Exercise 5.3 should be V [X] = ( p)/p. Page 474, 5.7: Answer should read n n k= n k k 5

6 Page 476, 7.5: change 0.69 to 0.84 Page 476, 7.5 should read f Z (t) = e t/ πt, for t > 0. Page 476, 7.7 should read f(x) = Page 48, Last two lines of E.4 should be: x π e x/, for x > 0. Y Pois(). Thus p = ppois(6, ) = The desired probability is P (X 3) = P (X ) = pbinom(, 00, ) = Page 48, next to last line on the page should be: 30 k=4 ( ) 30 (/) k (/) 30 k = k Page 48, E.8(e) Should be P (X = ) = 3/6, not P (X = ) = 3/6. Page 48, E.0(e). First part of display should be f Y (x) = (/)f X (x/) = Page 48, E.0(g). The solution should be [ ] e X e x E = 6x( x) dx = X x Page 483, top display should be 0 f M (m) = 30 ( m 4 ) 9 m xe x dx = 6. Page 483, the first display line of solution E.4: The last integral should be x e e t dt, Page 484, solution E.5(b). The last expression of the display should be ( e x/ )( e y/3 ) Page 484, solutions to E.8. The 3rd line should be The 5th line should be f Y (y) = f X Y (x y) = y x ln(/y) dx =, for 0 < y <. f(x, y) f Y (y) =, for y < x <. x ln(/y) Page 485, E.9(h(ii)). Change the + sign before to an equal sign = 6