Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page

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1 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 1 Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Practice exam 1:9, 1:22, 1:29, 9:5, and 10:8 are defective in that none of the five answer choices is correct. For 10:8, revised answer choices are provided. Note the change in wording to practice exam 1:1 (page 1247), 1:14 (page 1250), 1:15 (page 1251), 1:19 (page 1252), 1:24 (page 1253), 1:25 (page 1253), 1:28 (page 1254), 3:5 (page 1270), 5:15 (page 1293), 6:6 (page 1300), 8:20 (page 1326), 10:8 (page 1343), and 11:30 (page 1359). [3/25/2014] On page 11, in the third displayed line of Section 1.4, change e t x to e t X. [6/12/2014] On page 23, in the solution to exercise 1.15, the second displayed line should be 0.16 = exp z 2 z 2 2 (ln 5 ln 2) = exp ln 2/5 = [8/27/2013] On page 30, at the end of the second line of the solution to Example 2A, add be after will. [9/29/2013] On page 47, in the solution to exercise 2.1, on the last line, a 1 is missing from the numerator of the fraction. The numerator should be (4/3) [1/31/2014] On page 53, two lines above Continuity correction, change call to called. [10/18/2014] On page 91, in the solution to exercise 5.9, on the last line, change 2000 to [6/27/2014] On page 97, on the first line of the answer to Example 6F, the first sentence is inaccurate; E[(X 500) + ] is not calculated previously. However, it is easy to calculate: E[(X 500) + ] = 0.2( ) + 0.1( ) = 250. [2/22/2014] On page 97, 2 lines after the answer to Example 6H, change θ d to θ = d. [3/27/2014] On page 101, in exercise 6.4, in the second bullet, change r % to 100r %. [3/27/2014] On page 110, in the solution to exercise 6.2, on the second displayed line, place a parenthesis around [10/14/2013] On page 147, in Table 83, in item #3 under Tail Weight Measures, change heavier tail to lighter tail. z 2 /2 20,000 20,000 + d 2. [2/5/2014] On page 163, in exercise 9.13, although the exercise can be solved mechanically, the values in the table are impossible if losses are nonnegative. If F (100) = 0.3, then E[X 100] (1 0.3)(100) = 70. [4/1/2014] On page 169, in the solution to exercise 9.4, on the first line, add is after it. [6/2/2014] On page 174, in the solution to exercise 9.19, on the last displayed line, there should be a 0.8 on the left hand side, so that the part preceding the first equals sign is 0.8(E[X 5000] E[X 100]) [6/30/2013] On page 197, in the solution to exercise 11.1, the answer key should be (B) rather than (D). [7/11/2013] On page 198, in the solution to exercise 11.4, on the first line, change a = λ and b = 0 to a = 0 and b = λ. [9/10/2014] On page 204, in the solution to exercise 11.26, on the third line, add 1 plus before the second mean :... so its mean is 1 plus the mean of an unshifted.... [4/1/2014] On page 237, in the warning box on the second line, change variance distribution to variance formula. [8/27/2013] On page 238, in exercise 14.3(v), on the second line, change the period in ,000 to a comma. [8/1/2014] On page 273, in the solution to exercise 15.29, on the third line from the end, change to

2 2 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page [6/17/2013] On page 286, in the solution to exercise 16.7, on the fourth displayed line, delete the first F, so that the left side reads 50,000 1 F (50,000). [7/3/2014] On page 331, in the solution to exercise 19.1, on the last line, change 50 to 40. [6/17/2014] On page 332, in the solution to exercise 19.3, on the last line, change X > 3000 to S > [4/18/2014] On page 339, in exercise 20.11, delete the first line with s. [6/23/2014] On page 359, in exercise 21.9, on the sixth line, replace MSE β (ψ)/ MSE β (φ) with MSE ψ (β)/ MSE φ (β). [6/23/2014] On page 363, in the solution to exercise 21.9, on the first displayed line, replace MSE β (ψ) with MSE ψ (β). On the second displayed line, replace M S E β (φ) with MSE φ (β). [7/29/2014] On page 380, in exercise 23.11, on the last line, delete the hat on top of v (θ ). [6/18/2014] On page 384, in the solution to exercise 23.11, on the second line, change θ n is binomial to θ n is a binomial proportion. [7/29/2014] On page 384, in the solution to exercise 23.13, on the fifth line, change q = 1 7 to q = 6 7. [6/19/2014] On page 401, in exercise 24.29, on the fourth line, change states to times. [4/23/2014] On page 470, in the solution to exercise 27.9, on the third displayed line, change F n (17) = F n (16) to F n (17) F n (16). [8/5/2013] On page 480, in the second table, on the first line of the heading, change both Age 64 s to Age 65. [8/31/2013] On page 481, one line after equation (28.4), change the line to: Since ˆq j = nd j /e j, this can also be written as d j (e j /n d j ) (e j /n) 3. In particular, if n = 1 (one-year mortality rate, the most common case), then Var( ˆq j ) = d j (e j d j ). [8/5/2013] On page 481, one line above Section 28.2, the displayed line should be changed to e 3 j (4/9)(5/9) 54/12 = [8/31/2013] On page 487, change equation (28.2) to ˆq j = nd j e j. [6/11/2014] On page 493, in the solution to exercise 28.5, on the second line, replace toal with total. [2/2/2014] On page 494, in the solution to exercise 28.8, replace the top 3 lines of the page with Total exposure in months is 24(36) + 41(34) (29) (25) = The estimate of 3 q 25, by formula (28.2), is 3q 25 = nd j = 3(1) = 3/(3215/12) = e j 3215/12 The standard deviation, by formula (28.4), is Var( q j (1 q j ) ( )( ) 3 ˆq 25 ) = = = e j /n (3215/12)/3

3 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 3 [6/11/2014] On page 494, in the solution to exercise 28.9, on the fifth line, replace (33-5,38-5,x ) with (33.5,38-5,s ). [8/12/2013] On page 494, the solution to exercise is incorrect. The correct solution is P 41 = = 254. We assume uniform entry and withdrawal, so exact exposure is (8 10 2) = 252. The mortality estimate is 1 e 2/252 = [8/12/2013] On pages , the solution to exercise is incorrect. The correct solution is The sample mean is The second moment is = 6,000,000 5 Setting mean and second moment of lognormal equal to these, µ + σ2 = ln 2000 = µ + 2σ 2 = ln 6,000,000 = Double the first equation and subtract from the second. σ 2 = (7.6009) = µ = = ln Pr(X > 4500) = 1 Φ = 1 Φ(1.59) = = (B) [6/24/2014] On page 571, in exercise 32.15, in answer choice (B), the exponent 3 in the denominator should be outside the parenthesis, so that the denominator is f (100;θ ) 3. [5/8/2014] On page 618, in exercise 33.36, the correct solution, starting with the second line, is ˆα = n ln(1 + xi ) If the sample mean x i /n = x, then at least one of the x i s is greater than or equal to x. The other x i s are nonnegative. So n i =1 ln(1 + x i ) ln(1 + x ). As x, ln(1 + x ). Therefore ˆα 0. (A) [10/6/2013] On page 635, replace all lines after the S(10,000;α) line of the answer to Example 34E with: Let m be the the number of observations not exceeding 10,000. Dropping the +1 in the exponent of the denominator, which is a constant, the likelihood function is L(α) = α m (5500 8α ) m i =1 ( x i ) α (15,000 (8 m)α ) where the x i are the uncensored observations. We log and calculate the second derivative. m l (α) = m lnα + α 8 ln 5500 ln( x i ) (8 m) ln 15,000 dl dα = m α + 8 ln 5500 i =1 m ln( x i ) (8 m) ln 15,000 i =1

4 4 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page d 2 l dα 2 = m α 2 The expected value of m is 8 times the probability that an observation is less than 10,000 given that it is greater than 500, or 1 (5500/15,000) α, so E d2 l = 8(1 (5500/15,000)α ) dα 2 α 2 and the variance is the reciprocal of this expression. The variance can be estimated by using the estimate for α, obtained by setting the first derivative to 0 and setting the x i s equal to the 5 observations and m = 5: 5 (8 ln 5500 ln 6000 ln 7000 ln 8000 ln 11,000 ln 13,000 3 ln 15,000) = 0 α 5 ˆα = = Var( ˆα) = (1 (5500/15,000) ) = [8/4/2014] On page 634, 2 lines from the end of the page, change e 8250/10,000 to e 10,000/8250. [1/7/2014] On page 645, in exercise 34.2II, change Crarner to Cràmer. [6/26/2014] On page 655, in the solution to exercise 34.7, delete negative. [6/27/2014] On page 662, the solution to exercise is incorrect. The correct solution is The length of the 2-sided confidence interval for the proportion is 2 times 1.96 (the 97.5 th percentile of a standard normal distribution) times the standard deviation of the estimate for the proportion. We will use the delta method. In the previous exercise, we estimated ˆθ = 125. Let g (θ ) be Pr(X > 250 θ ). Then g (θ ) = S(250 θ ) = e 250/θ By the delta method Var g ( ˆθ ) Var( ˆθ ) g ( ˆθ ) 2 Since ˆθ is the sample mean, its variance is the variance of the distribution divided by the size of the sample. The variance of an exponential with mean θ is θ 2, so the variance of ˆθ is Var( ˆθ ) = θ 2 where as usual we approximate θ with its estimate. 100 g (θ ) = 250 θ 2 e 250/θ g ( ˆθ ) = e 250/125 = e 2 Var g ( ˆθ ) e 2 = 0.2e 2 2 The standard deviation of g ( ˆθ ) is 0.2e 2, and the length of the confidence interval is 2(1.96)(0.2e 2 ) = (E)

5 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 5 [6/30/2014] On page 745, in exercise 39.1, at the end of the last bullet, change good to fit to goodness of fit. [6/30/2014] On page 755, in exercise 39.23, after the table, add A Pareto distribution is fit to the data. [8/27/2013] On page 761, in the solution to exercise 39.6, on the sixth line, change 4041 to On the last line, change 4046 to 4047 in two places, change 990 to 900.and change the final answer to 258. [6/19/2014] On page 770, in the paragraph beginning Thus if an exponential model, on the fourth line, delete one that. [4/23/2014] On page 772, on the second-to-last line of the answer to Example 40C, replace 2( ) with 2( ). [8/4/2013] On page 787, in question 41.9, none of the answer choices is correct. [5/19/2014] On page 790, in the solution to exercise 41.5, on the first displayed line, change F ( x ) to F (x ). [8/4/2013] On page 792, in the solution to exercise 41.9, the solution, starting with the fourth line, should be corrected to the following: 3θ 3 Multiplying these together and logging, Multiplying out by θ 2 (θ + 260): (θ + 260) 4 l (θ ) = 3 lnθ lnθ 4 ln(θ + 260) + constant θ dl dθ = 891 θ 4 2 θ θ + 231,660 4θ 2 = 0 ˆθ = ,500,441 8 = [6/9/2014] On page 793, in the solution to exercise 41.13, in the table, change Transformed gamma to Generalized Pareto. [7/22/2013] On page 842, in the solution to exercise 44.2, on the third displayed line, the expression between the two equal signs should be (2000)(0.052 ). 3 [10/12/2014] On page 845, in the solution to exercise 44.18, replace the first line on the page with [9/16/2013] On page 895, delete exercise y 2 p = (0.12 ) = [5/28/2014] On page 963, exercise is a duplicate of exercise [8/30/2014] On pages , in the solution to exercise 53.17, replace the 9 lines starting with We want to calculate with We want to calculate Pr(1, 0 and p < 0.1) Pr(p < 0.1 1, 0) = Pr(1, 0)

6 6 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page Pr(1, 0 and p < 0.1) = Pr(1, 0 and p < 0.1) + Pr(1, 0 and p 0.1) We will obtain the numerator by integrating, from 0 to 0.1, Pr(1, 0) = p (1 p) over the density function of p. We will obtain the second term of the denominator by integrating the same integrand from 0.1 to 0.2. Replace the left side on the next line with Pr(1, 0 and p < 0.1. Replace the left side three lines after that line with Pr(1, 0 and p 0.1). Replace the final line of the solution (page 1035) with Pr(p < 0.1 1, 0) = 37/ / = = (B) [7/30/2013] On page 1051, in exercise 54.11, on the second to last line, change 2000 to [7/30/2013] On page 1056, the solution to exercise is incorrect. The correct solution is We will use one month as our exposure unit. Then µ = E[θ ] = 0.05 a = Var(θ ) = v = E[θ (1 θ )] = E[θ ] E[θ 2 ] = 0.05 ( ) = k = v a = There are 28 months of exposure and 1 claim per 28 months, so 28 Z = = P C = (1/28) + ( )(0.05) = Annualizing the estimate, we get 12( ) = [5/6/2014] On page 1069, in the solution to exercise 55.16, on the second line, change 0.2(0.5) + 0.8(3) to 0.8(0.5) + 0.2(3). [7/31/2013] On page 1089, the solution to exercise is incorrect. The correct solution is Bayesian and Bühlmann estimation are equivalent, so we can deduce Bühlmann credibility factors using the conditional expectation of X 2 given X 1. Since E[X 2 X 1 = 3] = 1.00, and 1.00 = 0.2(0.5) + 0.8(3), we see that the credibility factor for one observation is 0.2. Then 1/(1 + k) = 0.2 implying k = 4. The expected value of the process variance divided by the variance of the hypothetical means is 4. Then the expected value of the process variance is 4(6) = 24. (D) [6/26/2014] On page 1144, in the solution to exercise 59.16, on the second-to-last line, change 17,0460,223 to 17,046,223. [6/17/2014] On page 1150, on the first displayed line, change the integrand to [8/28/2013] On page 1154, in exercise 60.1, on the second line, change product to produce. On the third line, change y = 0.79 to u = u du [6/13/2014] On page 1159, exercise is a duplicate of exercise [8/28/2013] On page 1164, in the solution to exercise 60.1, on the second line, change ln u to ln(1 u).

7 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 7 [7/9/2014] On page 1167, in the solution to exercise 60.17, on the last line, change to [8/28/2013] On page 1173, replace the last three lines of the answer to Example 61A with The tables tell us that for a Pareto distribution, VaR p (Z ) = θ (1 p ) 1/α 1, so The average is (13, , )/3 = ( /3 1) = 200 [9/16/2013] On page 1175, 7 lines above Example 61C, change to On the next line, change to Note that the inverse of 0.99 appears in the normal distribution table at the bottom as 2.326, so perhaps should be used in the calculation on that line instead of However, the final answer is the same whether or 2.33 is used. [9/9/2013] On page 1176, in the paragraph after the table of the answer to Example 61C, on the second line, change it is at least equal to to it is less than. On the third line, change surrender in the first year to convert first year. [1/14/2014] On page 1176, two lines from the bottom, change to [8/1/2013] On page 1179, on the second line of Section 61.4, change µ + σx i to µ + σn i. In Example 61H on the first line, change a normal distribution with µ = 5 and σ 2 = 16 to a standard normal distribution. [8/1/2013] On page 1180, in Example 61I on the first line, change a normal distribution with µ = 5 and σ 2 = 16 to a standard normal distribution. [6/23/2014] On page 1185, in exercise 61.18, on the first line, change X (l ) to X (1). [9/24/2013] On page 1186, in the solution to exercise 61.3, on the last line, change 3 to 4. [7/16/2013] On page 1187, the solution to exercise 61.9 is incorrect. The correct solution is e 3 = (0.64)(0.5) = 0.32 (0.32)(0.25) = 0.08 (0.08)(0.75) = 0.06 (0.06)(0.50) = 0.03 It took 5 numbers to get below e 3, so we generate 4. (C) [12/3/2013] On page 1187, in the solution to exercise 61.10, on the last displayed line, change to [6/16/2014] On page 1189, in the solution to exercise 61.19, on the last line, change 2e to (2e ). [6/17/2014] On page 1213, in exercise 63.16, two lines after the enumerated list, change longer times to shorter times. [6/2/2014] On page 1226, 2 lines above Example 64F, change ˆσ 2 to s 2. [12/4/2013] On page 1247, in question 1, on the last line, change counts to costs. [6/10/2013] On page 1250, in question 14, delete the second-to-last line beginning The automobile collision. On the last line, change payments to losses. [6/10/2013] On page 1251(i), change x 15 to x 5. [6/11/2013] On page 1252, in question 19, on the fourth line, add 95% before log-transformed. [6/13/2013] On page 1253, in question 24, after the sentence under the table, Credibility... methods, add Annual claim counts for each individual are assumed to follow a Poisson distribution.

8 8 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page [9/18/2013] On page 1253, in question 25, on the second line, change 76 to 66. [6/11/2013] On page 1254, in question 28(iii), change 80% to 60%. [8/27/2013] On page 1270, in question 5, on the second row below the heading in the table (the one starting with 2 ), change the date of policy issue from 5/1/2010 to 5/1/2000. On the last line, change q 45 to q 35. [6/11/2013] On page 1293, in question 15, on the second line, change 720 to 240. [6/14/2013] On page 1300, in question 6, in the table, on the second line, change the date of policy issue from 6/1/2012 tgo 6/1/2002. [6/11/2013] On page 1326, in question 20, change the answer choices to (A) 19 (B) 20 (C) 21 (D) 22 (E) 23 [1/27/2014] On page 1343, question 8, replace 15 in the table with 19. Replace the answer choices with (A) 43.8 (B) 44.1 (C) 44.4 (D) 44.7 (E) 45.0 [6/21/2013] On page 1353, in question 10, on the last line, change the comma in to a period. [6/11/2013] On page 1359, in question 30, on the second line, delete negative. [6/10/2013] On page 1377, in the solution to question 7, on the second line, replace e θ x i with e θ 1/x i. On the third line, replace θ x i with θ 1 x i. [6/10/2013] On page 1377, in the solution to question 9, the last line should be None of the five choices is correct p 0 p 1 p 2 = 1 2 = ! 4 4 [8/12/2013] On page 1378, in the solution to question 11, on the sixth line, change e to e [8/12/2013] On page 1379, in the solution to question 13, on the fourth line, put a negative sign before and change x at the beginning of the line to a. [6/5/2014] On page 1379, in the solution to question 16, on the second line, in the exponent on (1 e 1000/θ ), change 94 to 84. [6/11/2013] On page 1381, in the solution to question 22, change the last three lines to None of the five choices is correct. a = 60( )2 + 20( ) ( )/80 k = = Z A = = (C) = [6/11/2013] On page 1382, in the solution to question 25, on the second line from the end, change to [9/18/2014] On page 1382, in the solution to question 26, on the first line, change 1000 to 1600 in two places. [6/11/2013] On page 1383, change the solution of question 28 to

9 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 9 Annual payments are 0.6(X 10,000 X 1,000), and we use the tables to evaluate the expected value of this. E[X 1,000] = θ α 1 θ 1 = = α 1 θ E[X 10,000] = = The answer is 0.6( ) = (A) [8/27/2013] On page 1383, the solution to question 29 is incorrect. The correct solution is Let q be the 95 th percentile. Let s calculate q. TVaR 0.95 (X ) = q + e (q ). We ll calculate e (q ) using 1 0.6e q /10 0.4e q /20 = x 2 q / x = 0.05 where x = e x = (0.6)(0.05) = q = 20 ln x = e (q ) = q x f (x ) dx = 0.05 q S(x ) dx The numerator is (0.6e x / e x /20 )dx = 10(0.6)e x /10 20(0.4)e x /20 = 6e q /10 + 8e q /20 = q q 0.05 So TVaR 0.95 (X ) = = (E) [6/11/2013] On page 1385, change the solution to question 35 to 1 plus the coefficient of variation squared can be expressed as For a Weibull distribution with τ = 0.5, 1 + C V 2 X = 1 + Var(X ) E[X ] = E[X ]2 + Var(X ) = E[X 2 ] 2 E[X ] 2 E[X ] 2 E[X ] = θ Γ (1 + 1/τ) = θ Γ (3) = 2θ E[X 2 ] = θ 2 Γ (1 + 2/τ) = θ 2 Γ (5) = 24θ 2 E[X 2 ] E[X ] = = 6 2 The standard for full credibility is λ F = (6) = (A) The answer key on page 1375 should also be corrected.

10 10 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page [6/11/2013] On page 1392, in the solution to question 17, on the third displayed line at the right, remove the negative sign in front of [3/22/2014] On page 1400, in the solution to question 5, change the two Age 45 s in the heading of the table to Age 35. Change the three subscripts 45 in the two lines after the table, and the subscript 40 in the displayed formula, to 35. [6/21/2013] On page 1402, in question 10, on the last line, change the comma in to a period. [6/11/2013] On page 1424, in the solution to question 2, on the last line, change the equals sign before to a minus sign. [6/11/2013] On page 1428, in the solution to question 15, replace the sentence starting 5 lines from the end beginning with For #6 with For #6, death occurs at age 29 so the scheduled 10 months of exposure are lost. Replace the last two sentences starting with So the total adjustment with So the total adjustment to actuarial exposure in months is = 22, or years. There are 2 deaths at age 30 (#3 and #7), so the actuarial estimate of q 30 is 2/( ) = [5/19/2014] On page 1461, the solution to question 8 has typos and is unclear. Here is a clearer solution: The function transforming X to Y is y = g (x ) = x 1 = 1/x. Notice that x = g 1 (y ) = 1/y. The derivative of g 1 (y ) is dg 1 dy = 1 y 2 By the formula for transforming densities of random variables, f Y (y ) = f X (1/y ) 1 = (1/(100y ))4 e 1/100y y 2 (1/y )Γ (4) 1 e 1/100y = Γ (4)y 5 1 Not surprisingly, inverting X results in an inverse gamma. If you check the tables, you will see that the new parameters are α = 4 and θ = 1/100. The mode of an inverse gamma according to the tables is θ /(α + 1) = 1/(100 5) = 1/500. (B) If you did not recognize the density as an inverse gamma, you could still find the mode by differentiating it and set the derivative equal to 0. It is easier to differentiate the log: ln f Y = 4 ln 100y 1 + ln y lnγ (4) 2 ln y 100y d ln f Y dy 5y = 0 y 2 = y y 2 1 y = 5 y y 2 = 0 y = [1/29/2014] On page 1470, in the solution to question 30, on the last line, interchange and 0.5.

11 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 11 [1/27/2014] On page 1473, in the solution to question 5, on the second-to-last line, replace with Replace the last line with 10 Var( 5 ˆq 60 ) = ( ) 2(5 ) 2 = The 95% confidence interval is ± 1.96( ) = (0.049,0.186). None of the answer choices is correct. [6/25/2013] On page 1485, in the solution to question 4, on the fifth line, delete given that it is greater than [1/27/2014] On page 1486, the solution to question 8 is incorrect. The correct solution is From the tabular values with d = 20. We also have E[X ] = E[X 20] + Pr(X > 20)e (20) = (200) = 199 E[X ] = E[X 50] + Pr(X > 50)e (50) Let s get a lower bound for Pr(X > 50) by using the values of E[X 50]. 20 Plugging this into (*), we get = E[X 50] Pr(X > 50) (*) E[X 20] = 20 Pr(X > 20) + x f (x ) dx = 1 19 = 20(0.9) + 20 E[X 50] = 50 Pr(X > 50) + = 50 Pr(X > 50) x f (x ) dx x f (x ) dx x f (x ) dx x f (x ) dx x f (x ) dx 50 Pr(X > 50) Pr(20 < X 50) = 50 Pr(X > 50) Pr(X > 50) = 30 Pr(X > 50) + 19 (**) Pr(X > 50) Pr(X > 50) = 210 Pr(X > 50) Pr(X > 50) 180 Pr(X > 50) 6 7 Then plugging into (**), 6 E[X 50] =

12 12 Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page To prove that this lower bound is attained, here is a discrete random variable satisfying the question s assumptions with E[X 50] = : x Pr(X = x ) [6/17/2014] On page 1508, in the solution to question 30, on the first line of the page, change to [6/11/2013] On page 1522, in the solution to question 28, on the third line from the end, change the second = to. [7/14/2013] On page 1581, in the solution to question 1, on the fourth line, change α1 F (u) to α 1 F (u). 6 7