Estimation, Evaluation, and Selection of Actuarial Models
Estimation, Evaluation, and Selection of Actuarial Models Estimation, Evaluation, and Selection of Actuarial Models
96 APPENDIX A. SOLUTIONS TO EXERCISES Graphs of the ogive and histogram appear below. F(x) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 500 1000 1500 2000 2500 3000 3500 4000 x f(x) 0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 0 0 500 1000 1500 2000 2500 3000 3500 4000 x Ogive for Exercise 2 Exercise 3 i d i u i x i i d i u i x i 1 0 - 0.1 16 0 4.8 - 2 0 - 0.5 17 0 - 4.8 3 0 - 0.8 18 0 - 4.8 4 0 0.8 - 19—30 0 - 5.0 5 0 - 1.8 31 0.3 - 5.0 6 0 - 1.8 32 0.7 - 5.0 7 0 - 2.1 33 1.0 4.1 - 8 0 - 2.5 34 1.8 3.1 - 9 0 - 2.8 35 2.1 - 3.9 10 0 2.9 - 36 2.9 - 5.0 11 0 2.9 - 37 2.9 - 4.8 12 0 - 3.9 38 3.2 4.0 - 13 0 4.0 - 39 3.4 - 5.0 14 0 - 4.0 40 3.9 - 5.0 15 0 - 4.1 j y j s j r j 1 0.1 1 30 − 0 − 0=30or 0+30− 0 − 0=30 2 0.5 1 31 − 1 − 0=30or 30 + 1 − 1 − 0=30 3 0.8 1 32 − 2 − 0=30or 30 + 1 − 1 − 0=30 4 1.8 2 33 − 3 − 1=29or 30 + 1 − 1 − 1=29 5 2.1 1 34 − 5 − 1=28or 29 + 1 − 2 − 0=28 6 2.5 1 35 − 6 − 1=28or 28 + 1 − 1 − 0=28 7 2.8 1 35 − 7 − 1=27or 28 + 0 − 1 − 0=27 8 3.9 2 39 − 8 − 4=27or 27 + 4 − 1 − 3=27 9 4.0 1 40 − 10 − 4=26or 27 + 1 − 2 − 0=26 10 4.1 1 40 − 11 − 6=23or 26 + 0 − 1 − 2=23 11 4.8 3 40 − 12 − 7=21or 23 + 0 − 1 − 1=21 12 5.0 17 40 − 15 − 8=17or 21 + 0 − 3 − 1=17 Histogram for Exercise 2
97 Exercise 4 ⎧ ⎪⎨ S 40 (t) = ⎪⎩ 1, 0 ≤ t
- Page 50 and 51: 46 CHAPTER 3. SAMPLING PROPERTIES O
- Page 52 and 53: 48 CHAPTER 3. SAMPLING PROPERTIES O
- Page 54 and 55: 50 CHAPTER 3. SAMPLING PROPERTIES O
- Page 56 and 57: 52 CHAPTER 3. SAMPLING PROPERTIES O
- Page 58 and 59: 54 CHAPTER 3. SAMPLING PROPERTIES O
- Page 60 and 61: 56 CHAPTER 3. SAMPLING PROPERTIES O
- Page 62 and 63: 58 CHAPTER 3. SAMPLING PROPERTIES O
- Page 64 and 65: 60 CHAPTER 3. SAMPLING PROPERTIES O
- Page 66 and 67: 62 CHAPTER 4. MODEL EVALUATION AND
- Page 68 and 69: 64 CHAPTER 4. MODEL EVALUATION AND
- Page 70 and 71: 66 CHAPTER 4. MODEL EVALUATION AND
- Page 72 and 73: 68 CHAPTER 4. MODEL EVALUATION AND
- Page 74 and 75: 70 CHAPTER 4. MODEL EVALUATION AND
- Page 76 and 77: 72 CHAPTER 4. MODEL EVALUATION AND
- Page 78 and 79: 74 CHAPTER 4. MODEL EVALUATION AND
- Page 80 and 81: 76 CHAPTER 4. MODEL EVALUATION AND
- Page 82 and 83: 78 CHAPTER 4. MODEL EVALUATION AND
- Page 84 and 85: 80 CHAPTER 4. MODEL EVALUATION AND
- Page 86 and 87: 82 CHAPTER 4. MODEL EVALUATION AND
- Page 88 and 89: 84 CHAPTER 4. MODEL EVALUATION AND
- Page 90 and 91: 86 CHAPTER 5. MODELS WITH COVARIATE
- Page 92 and 93: 88 CHAPTER 5. MODELS WITH COVARIATE
- Page 94 and 95: 90 CHAPTER 5. MODELS WITH COVARIATE
- Page 96 and 97: 92 CHAPTER 5. MODELS WITH COVARIATE
- Page 98 and 99: 94 CHAPTER 5. MODELS WITH COVARIATE
- Page 102 and 103: 98 APPENDIX A. SOLUTIONS TO EXERCIS
- Page 104 and 105: 100 APPENDIX A. SOLUTIONS TO EXERCI
- Page 106 and 107: 102 APPENDIX A. SOLUTIONS TO EXERCI
- Page 108 and 109: 104 APPENDIX A. SOLUTIONS TO EXERCI
- Page 110 and 111: 106 APPENDIX A. SOLUTIONS TO EXERCI
- Page 112 and 113: 108 APPENDIX A. SOLUTIONS TO EXERCI
- Page 114 and 115: 110 APPENDIX A. SOLUTIONS TO EXERCI
- Page 116 and 117: 112 APPENDIX A. SOLUTIONS TO EXERCI
- Page 118 and 119: 114 APPENDIX A. SOLUTIONS TO EXERCI
- Page 120 and 121: 116 APPENDIX A. SOLUTIONS TO EXERCI
- Page 122 and 123: 118 APPENDIX A. SOLUTIONS TO EXERCI
- Page 124 and 125: 120 APPENDIX A. SOLUTIONS TO EXERCI
- Page 126 and 127: 122 APPENDIX A. SOLUTIONS TO EXERCI
- Page 128 and 129: 124 APPENDIX A. SOLUTIONS TO EXERCI
- Page 130 and 131: 126 APPENDIX A. SOLUTIONS TO EXERCI
- Page 132 and 133: 128 APPENDIX A. SOLUTIONS TO EXERCI
- Page 134 and 135: 130 APPENDIX A. SOLUTIONS TO EXERCI
- Page 136 and 137: 132 APPENDIX B. USING MICROSOFT EXC
- Page 138 and 139: 134 APPENDIX B. USING MICROSOFT EXC
- Page 140 and 141: 136 APPENDIX B. USING MICROSOFT EXC
- Page 142: 138 APPENDIX B. USING MICROSOFT EXC
96 APPENDIX A. SOLUTIONS TO EXERCISES<br />
Graphs <strong>of</strong> the ogive <strong>and</strong> histogram appear below.<br />
F(x)<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
0 500 1000 1500 2000 2500 3000 3500 4000<br />
x<br />
f(x)<br />
0.004<br />
0.0035<br />
0.003<br />
0.0025<br />
0.002<br />
0.0015<br />
0.001<br />
0.0005<br />
0<br />
0 500 1000 1500 2000 2500 3000 3500 4000<br />
x<br />
Ogive for Exercise 2<br />
Exercise 3<br />
i d i u i x i i d i u i x i<br />
1 0 - 0.1 16 0 4.8 -<br />
2 0 - 0.5 17 0 - 4.8<br />
3 0 - 0.8 18 0 - 4.8<br />
4 0 0.8 - 19—30 0 - 5.0<br />
5 0 - 1.8 31 0.3 - 5.0<br />
6 0 - 1.8 32 0.7 - 5.0<br />
7 0 - 2.1 33 1.0 4.1 -<br />
8 0 - 2.5 34 1.8 3.1 -<br />
9 0 - 2.8 35 2.1 - 3.9<br />
10 0 2.9 - 36 2.9 - 5.0<br />
11 0 2.9 - 37 2.9 - 4.8<br />
12 0 - 3.9 38 3.2 4.0 -<br />
13 0 4.0 - 39 3.4 - 5.0<br />
14 0 - 4.0 40 3.9 - 5.0<br />
15 0 - 4.1<br />
j y j s j r j<br />
1 0.1 1 30 − 0 − 0=30or 0+30− 0 − 0=30<br />
2 0.5 1 31 − 1 − 0=30or 30 + 1 − 1 − 0=30<br />
3 0.8 1 32 − 2 − 0=30or 30 + 1 − 1 − 0=30<br />
4 1.8 2 33 − 3 − 1=29or 30 + 1 − 1 − 1=29<br />
5 2.1 1 34 − 5 − 1=28or 29 + 1 − 2 − 0=28<br />
6 2.5 1 35 − 6 − 1=28or 28 + 1 − 1 − 0=28<br />
7 2.8 1 35 − 7 − 1=27or 28 + 0 − 1 − 0=27<br />
8 3.9 2 39 − 8 − 4=27or 27 + 4 − 1 − 3=27<br />
9 4.0 1 40 − 10 − 4=26or 27 + 1 − 2 − 0=26<br />
10 4.1 1 40 − 11 − 6=23or 26 + 0 − 1 − 2=23<br />
11 4.8 3 40 − 12 − 7=21or 23 + 0 − 1 − 1=21<br />
12 5.0 17 40 − 15 − 8=17or 21 + 0 − 3 − 1=17<br />
Histogram for Exercise 2
Change language