(a) Use mortality given by table AMC00 ultimate values. You can find the table in the...

    

Transcribed Image Text:

(a) Use mortality given by table AMC00 ultimate values. You can find the table in the appendix. Determine the following quantities: (i) The probability that a man aged 25 dies during the next 10 years. (ii) The probability that a man aged 67 dies between the ages of 75 and 85. (iii) The expected number of survivors to age 60 from a group of 2,000 men aged 25. (b) Consider the survival function s(x)=1-, for 0x6. (i) Determine the probability density function for the complete lifetime of an individual of age 1. (ii) What is the probability that an individual aged 3 dies within the next 2 years? (iii) Calculate the complete expectation of a life at age 2 using this survival function. AMC00 Male Life Table Age 2 Iz -1+1 Age z Age r 17 9997.5094 18 10000.0000 9992.9305 9994.6901 9995.4200 17 59 Iz 9464.7096 9478.8531 9480.7305 -1+1 Age r 59 18 60 19 20 9988.3337 9990.1025 9990.8321 9983.6991 9985.4871 9986.2163 19 61 20 62 21 9979.0567 9980.8438 9981.5827 21 63 22 26 9960.0701 9961.9336 9955.1996 9957.0921 9974.3666 9976.1827 9976.9213 23 9969.6487 9971.4740 9972.2222 24 9964.8832 9966.7276 25 9967.4854 9962.7010 27 9950.2618 9952.1832 28 9945.2469 9947.1972 29 9957.8691 9952.9698 9947.9933 9940.1847 9942.1340 9942.9398 30 9935.0655 9936.9939 9937.7794 31 9929.8794 9931.7671 9932.5024 32 9924.5767 9926.4238 33 9919.1182 9920.9344 34 9913.4841 9915.2596 35 9907.6152 9909.3799 36 9901.4922 9903.2460 37 9895.0760 9896.8385 9927.0892 9921.5201 9915.7755 9909.8162 9903.6126 9897.1357 38 9888.3275 9890.1087 9890.3363 39 9881.1880 9882.9977 9883.1559 40 9873.5893 9875.4570 9875.5558 41 9865.4534 9867.4085 9867.4578 42 43 9856.7223 9858.7745 9858.7942 42 9847.3090 9849.4875 9849.4875 43 44 9837.1169 9839.4312 9839.4312 45 9826.0499 9828.5291 9828.5291 46 9813.9735 9816.6562 9816.6562 47 9800.7439 9803.6786 9803.6786 48 9786.2090 9789.4437 9789.4437 49 9770.1885 9773.7708 9773.7708 50 9752.4550 9756.4712 9756.4712 51 9732.8031 9737.3192 9737.3192 51 52 9710.9228 9716.0627 9716.0627 53 9686.5083 9692.4138 54 9659.2100 9666.0213 9692.4332 9666.1182 55 9628.6170 9636.5205 | 9636.7719 56 9594.3554 9603.4960 9604.0069 57 9555.9547 9566.5317 9567.3964 58 9512.9284 9525.1558 9526.4767 8838858885883885889=97799999858333353 22 64 23 65 24 66 25 67 26 27 68 69 28 29 70 71 30 72 31 73 32 34 76 77 36 78 79 40 41 44 45 46 84 85 86 87 88 9410.6849 9427.0212 9429.5820 60 9350.1805 9369.0144 9372.4010 61 9282.4873 9304.1589 9308.5187 62 9206.8320 9231.7028 9237.1968 63 9122.3956 9150.8360 9157.6369 64 9028.3257 9060.7100 9069.0001 65 8923.7129 8960.4327 8970.3747 66 8807.6361 8849.0571 8860.8106 67 8679.1539 8725.6105 8739.3111 68 8537.3365 8589.1250 8604.8568 69 8381.2654 8438.6022 8456.4058. 70 8210.1056 8273.0800 8292.9181 71 8023.0608 8091.6666 8113.3848 72 7819.4936 7893.5124 7916.8461 73 74 7598.8879 7677.9138 7702.4263 74 75 7360.9680 7444.3113 7469.3894 75 7105.6585 7192.3651 7217.1705 76 6833.2053 6921.9772 6945.4296 77 6543.2750 6633.3614 6654.1244 78 6234.6920 6326.2215 6343.5631 79 80 5905.8869 6000.3050 6014.4463 80 81 5557.1270 5655.4241 5667.9541 81 82 5191.0860 5293.5691 5305.7888 82 83 4811.5773 4917.5469 4930.2080 83 4422.4381 4531.0383 4544.0494 84 4027.2597 4137.4915 4150.7347 85 3630.1730 3740.8772 3754.2191 86 3235.7358 3345.6473 3358.9374 87 2848.8406 2956.5986 2969.6769 88 47 89 48 90 2474.5254 2578.7306 2591.4291 89 2117.7709 2217.0436 | 2229.1940 90 49 91 1876.3069 1887.7528 91 50 92 1571.4202 92 93 1283.8047 93 52 94 1027.5920 94 95 804.3661 95 54 96 614.5164 96 97 457.2229 97 56 98 99 100 330.5502 98 231.6268 99 156.9026 100 Age z Dr D-1+1 D Age r Age z 17 5132.45 5133.73 17 59 AMC00 Male Life Functions (4%) D D-1+1 D Age x 935.71 937.11 937.29 59 18 4932.79 4933.66 4934.02 18 19 4740.89 4741.73 4742.07 19 61 20 4556.43 4557.25 4557.58 20 62 815.83 21 22 4379.15 4379.93 4380.25 4208.74 4209.50 4209.82 22 21 63 64 741.27 743.58 23 4044.95 4045.69 4045.99 23 65 705.41 60 894.58 896.14 896.38 60 854.65 856.37 856.68 61 817.73 818.11 62 778.05 780.16 780.62 63 744.13 64 707.94 708.59 65 24 3887.51 3888.23 3888.53 24 66 670.42 673.18 673.92 66 25 3736.19 3736.89 3737.18 25 67 636.25 639.24 640.09 67 26 3590.73 3591.42 3591.70 26 68 602.85 606.08 607.03 68 27 3450.92 3451.58 3451.85 27 69 570.19 573.65 574.70 69 28 3316.52 3317.17 3317.43 28 70 538.24 541.92 543.07 70 29 3187.33 3187.96 3188.22 29 71 506.97 510.86 512.08 71 30 3063.17 3063.76 3064.00 30 72 476.36 480.44 481.73 72 31 2943.81 2944.37 2944.59 31 73 446.42 450.65 451.98 73 32 2829.08 2829.61 2829.80 32 74 417.14 421.48 422.82 74 33 2718.77 2719.27 2719.43 33 75 388.54 392.94 394.26 75 34 2612.72 2613.19 2613.32 34 76 360.64 365.04 366.30 76 35 2510.74 2511.19 2511.30 35 77 333.47 337.80 338.95 77 36 2412.68 2413.11 2413.20 36 78 307.04 311.27 312.24 78 37 2318.39 2318.80 2318.87 37 79 281.31 285.44 286.22 79 38 2227.70 2228.10 2228.15 38 80 256.22 260.32 260.93 80 39 2140.47 2140.86 2140.90 39 81 231.82 235.92 236.44 81 40 2056.56 2056.95 2056.97 40 82 208.22 212.33 212.82 82 41 42 1975.83 1976.22 1976.23 41 1898.16 1898.55 1898.56 42 83 185.58 189.66 190.15 83 84 164.01 168.03 43 1823.41 1823.81 1823.81 43 85 143.61 44 1751.46 1751.87 1751.87 44 86 45 1682.20 1682.63 1682.63 45 46 1615.52 1615.96 1615.96 46 87 88 47 48 49 1429.78 1430.30 1430.30 49 1551.29 1551.75 1551.75 47 1489.41 1489.90 1489.90 48 89 90 168.52 84 147.54 148.01 85 124.47 128.26 128.72 86 106.68 110.30 110.74 87 90.31 93.73 94.14 88 75.43 89 62.07 91 78.60 78.99 64.98 52.88 65.34 90 53.20 91 50 51 1372.29 1372.86 1372.86 50 1316.85 1317.47 1317.47 92 42.58 92 51 93 33.45 93 52 1263.36 1264.03 1264.03 52 94 25.74 94 53 1211.71 1212.45 |1212.46 53 95 19.38 95 54 1161.83 1162.65 1162.66 54 96 14.23 96 55 1113.60 1114.52 1114.55 55 97 10.18 97 56 1066.96 1067.98 1068.03 56 98 7.08 98 57 1021.82 1022.95 1023.04 57 99 4.77 99 58 978.09 979.35 979.49 58 100 3.11 100 AMC00 Male Life Functions (4%) Age r N N-1+1 NT Age r Age x N-1+1 17 121013.22 18 121014.86 115879.55 115880.77 115881.13 17 59 NE 14258.53 14260.18 14260.36 N Age r 59 18 60 13320.96 13322.83 13323.07 60 19 110945.59 110946.76 110947.11 19 61 12424.28 12426.38 12426.69 61 20 106203.56 106204.70 106205.03 20 62 21 101646.03 101647.13 101647.45 21 63 22 97265.82 97266.89 97267.20 22 64 11567.26 11569.63 11570.01 10748.78 10751.44 10751.90 9967.77 9970.73 9971.28 64 62 63 23 93056.05 93057.08 93057.39 23 65 9223.22 24 89010.09 89011.10 89011.39 24 66 8514.21 9226.50 9227.15 8517.82 8518.56 65 66 25 85121.59 85122.57 85122.86 25 67 7839.85 7843.79 7844.64 67 26 81384.45 81385.41 81385.69 26 68 7199.32 7203.60 7204.55 68 27 77792.78 77793.72 77793.99 27 69 6591.87 6596.47 6597.52 69 28 74340.96 74341.87 74342.13 28 70 6016.77 29 71023.58 71024.44 71024.70 29 71 5473.35 30 67835.43 67836.24 67836.49 30 72 4960.98 31 32 64771.51 64772.26 61827.01 61827.70 64772.48 31 73 4479.04 6021.68 6022.82 70 5478.53 5479.75 71 4966.38 4967.67 72 4484.61 4485.94 73 61827.89 32 74 4026.96 4032.62 4033.96 74 33 58997.30 58997.93 58998.09 33 75 3604.16 3609.81 3611.14 75 34 56277.95 56278.53 56278.66 34 76 3210.07 3215.62 3216.88 76 35 53664.69 53665.23 53665.34 35 77 2844.13 2849.44 2850.58 77 36 51153.45 51153.95 51154.04 36 78 2505.65 2510.66 2511.63 78 37 48740.30 48740.77 48740.84 37 79 2193.87 2198.61 2199.39 79 38 46421.48 46421.92 46421.97 38 80 1907.94 1912.56 1913.17 80 39 44193.38 44193.79 44193.82 39 81 1647.13 1651.72 1652.24 81 40 41 42 42052.51 42052.91 42052.93 39995.55 39995.95 39995.96 38019.32 38019.72 40 82 1410.71 1415.31 1415.80 82 41 83 1197.92 1202.49 1202.97 83 38019.72 42 84 1007.84 1012.34 1012.82 84 43 36120.76 36121.17 36121.17 43 85 839.45 843.83 844.31 85 44 34296.94 34297.35 34297.35 44 86 691.61 695.84 696.30 86 45 32545.06 32545.48 32545.48 45 87 563.10 567.14 567.58 87 46 30862.41 30862.85 30862.85 46 88 452.62 456.42 456.84 88 47 29246.43 29246.89 29246.89 47 89 358.78 362.31 362.70 89 48 27694.65 27695.14 27695.14 48 90 280.12 283.35 283.71 90 49 26204.72 26205.24 26205.24 49 91 218.05 218.37 91 50 24774.37 24774.94 24774.94 50 92 165.17 92 51 23401.47 23402.08 23402.08 51 93 122.59 93 52 22083.94 22084.61 22084.61 52 94 89.14 94 53 20819.83 20820.58 54 19607.27 19608.12 20820.58 53 95 63.39 95 19608.13 54 96 44.02 96 55 18444.47 18445.44 18445.47 55 97 29.78 97 56 17329.76 17330.87 17330.93 56 98 19.60 98 57 16261.53 16262.80 16262.89 57 99 12.52 99 58 15238.27 15239.71 15239.85 58 100 7.75 100 (a) Use mortality given by table AMC00 ultimate values. You can find the table in the appendix. Determine the following quantities: (i) The probability that a man aged 25 dies during the next 10 years. (ii) The probability that a man aged 67 dies between the ages of 75 and 85. (iii) The expected number of survivors to age 60 from a group of 2,000 men aged 25. (b) Consider the survival function s(x)=1-, for 0x6. (i) Determine the probability density function for the complete lifetime of an individual of age 1. (ii) What is the probability that an individual aged 3 dies within the next 2 years? (iii) Calculate the complete expectation of a life at age 2 using this survival function. AMC00 Male Life Table Age 2 Iz -1+1 Age z Age r 17 9997.5094 18 10000.0000 9992.9305 9994.6901 9995.4200 17 59 Iz 9464.7096 9478.8531 9480.7305 -1+1 Age r 59 18 60 19 20 9988.3337 9990.1025 9990.8321 9983.6991 9985.4871 9986.2163 19 61 20 62 21 9979.0567 9980.8438 9981.5827 21 63 22 26 9960.0701 9961.9336 9955.1996 9957.0921 9974.3666 9976.1827 9976.9213 23 9969.6487 9971.4740 9972.2222 24 9964.8832 9966.7276 25 9967.4854 9962.7010 27 9950.2618 9952.1832 28 9945.2469 9947.1972 29 9957.8691 9952.9698 9947.9933 9940.1847 9942.1340 9942.9398 30 9935.0655 9936.9939 9937.7794 31 9929.8794 9931.7671 9932.5024 32 9924.5767 9926.4238 33 9919.1182 9920.9344 34 9913.4841 9915.2596 35 9907.6152 9909.3799 36 9901.4922 9903.2460 37 9895.0760 9896.8385 9927.0892 9921.5201 9915.7755 9909.8162 9903.6126 9897.1357 38 9888.3275 9890.1087 9890.3363 39 9881.1880 9882.9977 9883.1559 40 9873.5893 9875.4570 9875.5558 41 9865.4534 9867.4085 9867.4578 42 43 9856.7223 9858.7745 9858.7942 42 9847.3090 9849.4875 9849.4875 43 44 9837.1169 9839.4312 9839.4312 45 9826.0499 9828.5291 9828.5291 46 9813.9735 9816.6562 9816.6562 47 9800.7439 9803.6786 9803.6786 48 9786.2090 9789.4437 9789.4437 49 9770.1885 9773.7708 9773.7708 50 9752.4550 9756.4712 9756.4712 51 9732.8031 9737.3192 9737.3192 51 52 9710.9228 9716.0627 9716.0627 53 9686.5083 9692.4138 54 9659.2100 9666.0213 9692.4332 9666.1182 55 9628.6170 9636.5205 | 9636.7719 56 9594.3554 9603.4960 9604.0069 57 9555.9547 9566.5317 9567.3964 58 9512.9284 9525.1558 9526.4767 8838858885883885889=97799999858333353 22 64 23 65 24 66 25 67 26 27 68 69 28 29 70 71 30 72 31 73 32 34 76 77 36 78 79 40 41 44 45 46 84 85 86 87 88 9410.6849 9427.0212 9429.5820 60 9350.1805 9369.0144 9372.4010 61 9282.4873 9304.1589 9308.5187 62 9206.8320 9231.7028 9237.1968 63 9122.3956 9150.8360 9157.6369 64 9028.3257 9060.7100 9069.0001 65 8923.7129 8960.4327 8970.3747 66 8807.6361 8849.0571 8860.8106 67 8679.1539 8725.6105 8739.3111 68 8537.3365 8589.1250 8604.8568 69 8381.2654 8438.6022 8456.4058. 70 8210.1056 8273.0800 8292.9181 71 8023.0608 8091.6666 8113.3848 72 7819.4936 7893.5124 7916.8461 73 74 7598.8879 7677.9138 7702.4263 74 75 7360.9680 7444.3113 7469.3894 75 7105.6585 7192.3651 7217.1705 76 6833.2053 6921.9772 6945.4296 77 6543.2750 6633.3614 6654.1244 78 6234.6920 6326.2215 6343.5631 79 80 5905.8869 6000.3050 6014.4463 80 81 5557.1270 5655.4241 5667.9541 81 82 5191.0860 5293.5691 5305.7888 82 83 4811.5773 4917.5469 4930.2080 83 4422.4381 4531.0383 4544.0494 84 4027.2597 4137.4915 4150.7347 85 3630.1730 3740.8772 3754.2191 86 3235.7358 3345.6473 3358.9374 87 2848.8406 2956.5986 2969.6769 88 47 89 48 90 2474.5254 2578.7306 2591.4291 89 2117.7709 2217.0436 | 2229.1940 90 49 91 1876.3069 1887.7528 91 50 92 1571.4202 92 93 1283.8047 93 52 94 1027.5920 94 95 804.3661 95 54 96 614.5164 96 97 457.2229 97 56 98 99 100 330.5502 98 231.6268 99 156.9026 100 Age z Dr D-1+1 D Age r Age z 17 5132.45 5133.73 17 59 AMC00 Male Life Functions (4%) D D-1+1 D Age x 935.71 937.11 937.29 59 18 4932.79 4933.66 4934.02 18 19 4740.89 4741.73 4742.07 19 61 20 4556.43 4557.25 4557.58 20 62 815.83 21 22 4379.15 4379.93 4380.25 4208.74 4209.50 4209.82 22 21 63 64 741.27 743.58 23 4044.95 4045.69 4045.99 23 65 705.41 60 894.58 896.14 896.38 60 854.65 856.37 856.68 61 817.73 818.11 62 778.05 780.16 780.62 63 744.13 64 707.94 708.59 65 24 3887.51 3888.23 3888.53 24 66 670.42 673.18 673.92 66 25 3736.19 3736.89 3737.18 25 67 636.25 639.24 640.09 67 26 3590.73 3591.42 3591.70 26 68 602.85 606.08 607.03 68 27 3450.92 3451.58 3451.85 27 69 570.19 573.65 574.70 69 28 3316.52 3317.17 3317.43 28 70 538.24 541.92 543.07 70 29 3187.33 3187.96 3188.22 29 71 506.97 510.86 512.08 71 30 3063.17 3063.76 3064.00 30 72 476.36 480.44 481.73 72 31 2943.81 2944.37 2944.59 31 73 446.42 450.65 451.98 73 32 2829.08 2829.61 2829.80 32 74 417.14 421.48 422.82 74 33 2718.77 2719.27 2719.43 33 75 388.54 392.94 394.26 75 34 2612.72 2613.19 2613.32 34 76 360.64 365.04 366.30 76 35 2510.74 2511.19 2511.30 35 77 333.47 337.80 338.95 77 36 2412.68 2413.11 2413.20 36 78 307.04 311.27 312.24 78 37 2318.39 2318.80 2318.87 37 79 281.31 285.44 286.22 79 38 2227.70 2228.10 2228.15 38 80 256.22 260.32 260.93 80 39 2140.47 2140.86 2140.90 39 81 231.82 235.92 236.44 81 40 2056.56 2056.95 2056.97 40 82 208.22 212.33 212.82 82 41 42 1975.83 1976.22 1976.23 41 1898.16 1898.55 1898.56 42 83 185.58 189.66 190.15 83 84 164.01 168.03 43 1823.41 1823.81 1823.81 43 85 143.61 44 1751.46 1751.87 1751.87 44 86 45 1682.20 1682.63 1682.63 45 46 1615.52 1615.96 1615.96 46 87 88 47 48 49 1429.78 1430.30 1430.30 49 1551.29 1551.75 1551.75 47 1489.41 1489.90 1489.90 48 89 90 168.52 84 147.54 148.01 85 124.47 128.26 128.72 86 106.68 110.30 110.74 87 90.31 93.73 94.14 88 75.43 89 62.07 91 78.60 78.99 64.98 52.88 65.34 90 53.20 91 50 51 1372.29 1372.86 1372.86 50 1316.85 1317.47 1317.47 92 42.58 92 51 93 33.45 93 52 1263.36 1264.03 1264.03 52 94 25.74 94 53 1211.71 1212.45 |1212.46 53 95 19.38 95 54 1161.83 1162.65 1162.66 54 96 14.23 96 55 1113.60 1114.52 1114.55 55 97 10.18 97 56 1066.96 1067.98 1068.03 56 98 7.08 98 57 1021.82 1022.95 1023.04 57 99 4.77 99 58 978.09 979.35 979.49 58 100 3.11 100 AMC00 Male Life Functions (4%) Age r N N-1+1 NT Age r Age x N-1+1 17 121013.22 18 121014.86 115879.55 115880.77 115881.13 17 59 NE 14258.53 14260.18 14260.36 N Age r 59 18 60 13320.96 13322.83 13323.07 60 19 110945.59 110946.76 110947.11 19 61 12424.28 12426.38 12426.69 61 20 106203.56 106204.70 106205.03 20 62 21 101646.03 101647.13 101647.45 21 63 22 97265.82 97266.89 97267.20 22 64 11567.26 11569.63 11570.01 10748.78 10751.44 10751.90 9967.77 9970.73 9971.28 64 62 63 23 93056.05 93057.08 93057.39 23 65 9223.22 24 89010.09 89011.10 89011.39 24 66 8514.21 9226.50 9227.15 8517.82 8518.56 65 66 25 85121.59 85122.57 85122.86 25 67 7839.85 7843.79 7844.64 67 26 81384.45 81385.41 81385.69 26 68 7199.32 7203.60 7204.55 68 27 77792.78 77793.72 77793.99 27 69 6591.87 6596.47 6597.52 69 28 74340.96 74341.87 74342.13 28 70 6016.77 29 71023.58 71024.44 71024.70 29 71 5473.35 30 67835.43 67836.24 67836.49 30 72 4960.98 31 32 64771.51 64772.26 61827.01 61827.70 64772.48 31 73 4479.04 6021.68 6022.82 70 5478.53 5479.75 71 4966.38 4967.67 72 4484.61 4485.94 73 61827.89 32 74 4026.96 4032.62 4033.96 74 33 58997.30 58997.93 58998.09 33 75 3604.16 3609.81 3611.14 75 34 56277.95 56278.53 56278.66 34 76 3210.07 3215.62 3216.88 76 35 53664.69 53665.23 53665.34 35 77 2844.13 2849.44 2850.58 77 36 51153.45 51153.95 51154.04 36 78 2505.65 2510.66 2511.63 78 37 48740.30 48740.77 48740.84 37 79 2193.87 2198.61 2199.39 79 38 46421.48 46421.92 46421.97 38 80 1907.94 1912.56 1913.17 80 39 44193.38 44193.79 44193.82 39 81 1647.13 1651.72 1652.24 81 40 41 42 42052.51 42052.91 42052.93 39995.55 39995.95 39995.96 38019.32 38019.72 40 82 1410.71 1415.31 1415.80 82 41 83 1197.92 1202.49 1202.97 83 38019.72 42 84 1007.84 1012.34 1012.82 84 43 36120.76 36121.17 36121.17 43 85 839.45 843.83 844.31 85 44 34296.94 34297.35 34297.35 44 86 691.61 695.84 696.30 86 45 32545.06 32545.48 32545.48 45 87 563.10 567.14 567.58 87 46 30862.41 30862.85 30862.85 46 88 452.62 456.42 456.84 88 47 29246.43 29246.89 29246.89 47 89 358.78 362.31 362.70 89 48 27694.65 27695.14 27695.14 48 90 280.12 283.35 283.71 90 49 26204.72 26205.24 26205.24 49 91 218.05 218.37 91 50 24774.37 24774.94 24774.94 50 92 165.17 92 51 23401.47 23402.08 23402.08 51 93 122.59 93 52 22083.94 22084.61 22084.61 52 94 89.14 94 53 20819.83 20820.58 54 19607.27 19608.12 20820.58 53 95 63.39 95 19608.13 54 96 44.02 96 55 18444.47 18445.44 18445.47 55 97 29.78 97 56 17329.76 17330.87 17330.93 56 98 19.60 98 57 16261.53 16262.80 16262.89 57 99 12.52 99 58 15238.27 15239.71 15239.85 58 100 7.75 100