#2.6.3.a
contoh.2.6.3 <- function(expr, omega, l0) {
age <- 0:omega
sx <- eval({x=age; expr})
lx <- l0*sx
dx=px=qx <- NULL
n <- length(age+1)
for (i in 1:n) {
dx[i] <- lx[i]-lx[i+1]
px[i] <- lx[i+1]/lx[i]
qx[i] <- 1-lx[i+1]/lx[i] }
ex <- NULL
for (i in 1:n) {
ex[n] <- 0
ex[n-i] <- px[n-i]+(px[n-i]*ex[n-i+1])
}
tabel <- data.frame(age=age, qx=qx, px=px, lx=lx, dx=dx, sx=sx, ex=ex)
print(tabel, digits=3)
}
# contoh.2.6.3 b
expr <- expression(1-(0.01*x)^2)
contoh.2.6.3(expr, omega=99, l0=100000)
## age qx px lx dx sx ex
## 1 0 0.000100 1.000 100000 10 1.0000 66.165
## 2 1 0.000300 1.000 99990 30 0.9999 65.172
## 3 2 0.000500 0.999 99960 50 0.9996 64.191
## 4 3 0.000701 0.999 99910 70 0.9991 63.223
## 5 4 0.000901 0.999 99840 90 0.9984 62.268
## 6 5 0.001103 0.999 99750 110 0.9975 61.324
## 7 6 0.001305 0.999 99640 130 0.9964 60.392
## 8 7 0.001507 0.998 99510 150 0.9951 59.470
## 9 8 0.001711 0.998 99360 170 0.9936 58.560
## 10 9 0.001916 0.998 99190 190 0.9919 57.661
## 11 10 0.002121 0.998 99000 210 0.9900 56.771
## 12 11 0.002328 0.998 98790 230 0.9879 55.892
## 13 12 0.002537 0.997 98560 250 0.9856 55.022
## 14 13 0.002746 0.997 98310 270 0.9831 54.162
## 15 14 0.002958 0.997 98040 290 0.9804 53.311
## 16 15 0.003171 0.997 97750 310 0.9775 52.470
## 17 16 0.003387 0.997 97440 330 0.9744 51.636
## 18 17 0.003604 0.996 97110 350 0.9711 50.812
## 19 18 0.003824 0.996 96760 370 0.9676 49.996
## 20 19 0.004046 0.996 96390 390 0.9639 49.188
## 21 20 0.004271 0.996 96000 410 0.9600 48.387
## 22 21 0.004498 0.996 95590 430 0.9559 47.595
## 23 22 0.004729 0.995 95160 450 0.9516 46.810
## 24 23 0.004963 0.995 94710 470 0.9471 46.033
## 25 24 0.005199 0.995 94240 490 0.9424 45.262
## 26 25 0.005440 0.995 93750 510 0.9375 44.499
## 27 26 0.005684 0.994 93240 530 0.9324 43.742
## 28 27 0.005932 0.994 92710 550 0.9271 42.992
## 29 28 0.006185 0.994 92160 570 0.9216 42.249
## 30 29 0.006442 0.994 91590 590 0.9159 41.512
## 31 30 0.006703 0.993 91000 610 0.9100 40.781
## 32 31 0.006970 0.993 90390 630 0.9039 40.056
## 33 32 0.007242 0.993 89760 650 0.8976 39.337
## 34 33 0.007519 0.992 89110 670 0.8911 38.624
## 35 34 0.007802 0.992 88440 690 0.8844 37.917
## 36 35 0.008091 0.992 87750 710 0.8775 37.215
## 37 36 0.008387 0.992 87040 730 0.8704 36.518
## 38 37 0.008690 0.991 86310 750 0.8631 35.827
## 39 38 0.009000 0.991 85560 770 0.8556 35.141
## 40 39 0.009317 0.991 84790 790 0.8479 34.460
## 41 40 0.009643 0.990 84000 810 0.8400 33.785
## 42 41 0.009977 0.990 83190 830 0.8319 33.113
## 43 42 0.010321 0.990 82360 850 0.8236 32.447
## 44 43 0.010674 0.989 81510 870 0.8151 31.786
## 45 44 0.011037 0.989 80640 890 0.8064 31.128
## 46 45 0.011411 0.989 79750 910 0.7975 30.476
## 47 46 0.011796 0.988 78840 930 0.7884 29.828
## 48 47 0.012194 0.988 77910 950 0.7791 29.184
## 49 48 0.012604 0.987 76960 970 0.7696 28.544
## 50 49 0.013028 0.987 75990 990 0.7599 27.908
## 51 50 0.013467 0.987 75000 1010 0.7500 27.277
## 52 51 0.013921 0.986 73990 1030 0.7399 26.649
## 53 52 0.014391 0.986 72960 1050 0.7296 26.025
## 54 53 0.014880 0.985 71910 1070 0.7191 25.405
## 55 54 0.015387 0.985 70840 1090 0.7084 24.789
## 56 55 0.015914 0.984 69750 1110 0.6975 24.176
## 57 56 0.016463 0.984 68640 1130 0.6864 23.567
## 58 57 0.017035 0.983 67510 1150 0.6751 22.962
## 59 58 0.017631 0.982 66360 1170 0.6636 22.360
## 60 59 0.018254 0.982 65190 1190 0.6519 21.761
## 61 60 0.018906 0.981 64000 1210 0.6400 21.166
## 62 61 0.019589 0.980 62790 1230 0.6279 20.573
## 63 62 0.020305 0.980 61560 1250 0.6156 19.985
## 64 63 0.021058 0.979 60310 1270 0.6031 19.399
## 65 64 0.021850 0.978 59040 1290 0.5904 18.816
## 66 65 0.022684 0.977 57750 1310 0.5775 18.236
## 67 66 0.023565 0.976 56440 1330 0.5644 17.660
## 68 67 0.024496 0.976 55110 1350 0.5511 17.086
## 69 68 0.025484 0.975 53760 1370 0.5376 16.515
## 70 69 0.026532 0.973 52390 1390 0.5239 15.947
## 71 70 0.027647 0.972 51000 1410 0.5100 15.381
## 72 71 0.028836 0.971 49590 1430 0.4959 14.819
## 73 72 0.030108 0.970 48160 1450 0.4816 14.259
## 74 73 0.031471 0.969 46710 1470 0.4671 13.701
## 75 74 0.032935 0.967 45240 1490 0.4524 13.147
## 76 75 0.034514 0.965 43750 1510 0.4375 12.594
## 77 76 0.036222 0.964 42240 1530 0.4224 12.045
## 78 77 0.038074 0.962 40710 1550 0.4071 11.497
## 79 78 0.040092 0.960 39160 1570 0.3916 10.952
## 80 79 0.042298 0.958 37590 1590 0.3759 10.410
## 81 80 0.044722 0.955 36000 1610 0.3600 9.869
## 82 81 0.047397 0.953 34390 1630 0.3439 9.331
## 83 82 0.050366 0.950 32760 1650 0.3276 8.796
## 84 83 0.053680 0.946 31110 1670 0.3111 8.262
## 85 84 0.057405 0.943 29440 1690 0.2944 7.731
## 86 85 0.061622 0.938 27750 1710 0.2775 7.202
## 87 86 0.066436 0.934 26040 1730 0.2604 6.675
## 88 87 0.071987 0.928 24310 1750 0.2431 6.150
## 89 88 0.078457 0.922 22560 1770 0.2256 5.627
## 90 89 0.086099 0.914 20790 1790 0.2079 5.106
## 91 90 0.095263 0.905 19000 1810 0.1900 4.587
## 92 91 0.106457 0.894 17190 1830 0.1719 4.070
## 93 92 0.120443 0.880 15360 1850 0.1536 3.555
## 94 93 0.138416 0.862 13510 1870 0.1351 3.041
## 95 94 0.162371 0.838 11640 1890 0.1164 2.530
## 96 95 0.195897 0.804 9750 1910 0.0975 2.021
## 97 96 0.246173 0.754 7840 1930 0.0784 1.513
## 98 97 0.329949 0.670 5910 1950 0.0591 1.007
## 99 98 0.497475 0.503 3960 1970 0.0396 0.503
## 100 99 NA NA 1990 NA 0.0199 0.000
# contoh.2.6.3 b
expr <- expression(exp((-0.0001/log(1.09)*(1.09^x-1))))
contoh.2.6.3(expr, omega=99, l0=100000)
## age qx px lx dx sx ex
## 1 0 0.000104 1.000 100000 10.4 1.00000 71.326
## 2 1 0.000114 1.000 99990 11.4 0.99990 70.334
## 3 2 0.000124 1.000 99978 12.4 0.99978 69.342
## 4 3 0.000135 1.000 99966 13.5 0.99966 68.350
## 5 4 0.000147 1.000 99952 14.7 0.99952 67.360
## 6 5 0.000161 1.000 99938 16.1 0.99938 66.369
## 7 6 0.000175 1.000 99921 17.5 0.99921 65.380
## 8 7 0.000191 1.000 99904 19.1 0.99904 64.392
## 9 8 0.000208 1.000 99885 20.8 0.99885 63.404
## 10 9 0.000227 1.000 99864 22.6 0.99864 62.417
## 11 10 0.000247 1.000 99841 24.7 0.99841 61.431
## 12 11 0.000269 1.000 99817 26.9 0.99817 60.446
## 13 12 0.000294 1.000 99790 29.3 0.99790 59.463
## 14 13 0.000320 1.000 99761 31.9 0.99761 58.480
## 15 14 0.000349 1.000 99729 34.8 0.99729 57.499
## 16 15 0.000380 1.000 99694 37.9 0.99694 56.519
## 17 16 0.000415 1.000 99656 41.3 0.99656 55.540
## 18 17 0.000452 1.000 99615 45.0 0.99615 54.564
## 19 18 0.000493 1.000 99570 49.0 0.99570 53.588
## 20 19 0.000537 0.999 99521 53.4 0.99521 52.615
## 21 20 0.000585 0.999 99467 58.2 0.99467 51.643
## 22 21 0.000638 0.999 99409 63.4 0.99409 50.673
## 23 22 0.000695 0.999 99346 69.1 0.99346 49.705
## 24 23 0.000758 0.999 99276 75.2 0.99276 48.740
## 25 24 0.000826 0.999 99201 81.9 0.99201 47.777
## 26 25 0.000900 0.999 99119 89.2 0.99119 46.816
## 27 26 0.000981 0.999 99030 97.2 0.99030 45.859
## 28 27 0.001069 0.999 98933 105.8 0.98933 44.904
## 29 28 0.001166 0.999 98827 115.2 0.98827 43.952
## 30 29 0.001270 0.999 98712 125.4 0.98712 43.003
## 31 30 0.001385 0.999 98587 136.5 0.98587 42.058
## 32 31 0.001509 0.998 98450 148.6 0.98450 41.116
## 33 32 0.001645 0.998 98301 161.7 0.98301 40.178
## 34 33 0.001793 0.998 98140 175.9 0.98140 39.244
## 35 34 0.001954 0.998 97964 191.4 0.97964 38.315
## 36 35 0.002130 0.998 97772 208.2 0.97772 37.390
## 37 36 0.002321 0.998 97564 226.5 0.97564 36.470
## 38 37 0.002530 0.997 97338 246.2 0.97338 35.555
## 39 38 0.002757 0.997 97091 267.7 0.97091 34.645
## 40 39 0.003005 0.997 96824 290.9 0.96824 33.740
## 41 40 0.003275 0.997 96533 316.1 0.96533 32.842
## 42 41 0.003569 0.996 96217 343.4 0.96217 31.950
## 43 42 0.003890 0.996 95873 372.9 0.95873 31.065
## 44 43 0.004239 0.996 95500 404.8 0.95500 30.186
## 45 44 0.004620 0.995 95096 439.3 0.95096 29.314
## 46 45 0.005034 0.995 94656 476.5 0.94656 28.450
## 47 46 0.005486 0.995 94180 516.7 0.94180 27.594
## 48 47 0.005978 0.994 93663 560.0 0.93663 26.747
## 49 48 0.006515 0.993 93103 606.5 0.93103 25.907
## 50 49 0.007099 0.993 92497 656.6 0.92497 25.077
## 51 50 0.007735 0.992 91840 710.4 0.91840 24.257
## 52 51 0.008429 0.992 91129 768.1 0.91129 23.446
## 53 52 0.009184 0.991 90361 829.9 0.90361 22.645
## 54 53 0.010006 0.990 89531 895.9 0.89531 21.855
## 55 54 0.010902 0.989 88636 966.3 0.88636 21.076
## 56 55 0.011877 0.988 87669 1041.3 0.87669 20.308
## 57 56 0.012939 0.987 86628 1120.9 0.86628 19.552
## 58 57 0.014095 0.986 85507 1205.3 0.85507 18.809
## 59 58 0.015354 0.985 84302 1294.4 0.84302 18.077
## 60 59 0.016725 0.983 83008 1388.3 0.83008 17.359
## 61 60 0.018216 0.982 81619 1486.8 0.81619 16.655
## 62 61 0.019839 0.980 80132 1589.7 0.80132 15.964
## 63 62 0.021605 0.978 78543 1696.9 0.78543 15.287
## 64 63 0.023527 0.976 76846 1807.9 0.76846 14.624
## 65 64 0.025617 0.974 75038 1922.2 0.75038 13.977
## 66 65 0.027890 0.972 73116 2039.2 0.73116 13.344
## 67 66 0.030361 0.970 71077 2158.0 0.71077 12.727
## 68 67 0.033048 0.967 68919 2277.6 0.68919 12.125
## 69 68 0.035968 0.964 66641 2397.0 0.66641 11.540
## 70 69 0.039141 0.961 64244 2514.6 0.64244 10.970
## 71 70 0.042588 0.957 61729 2628.9 0.61729 10.417
## 72 71 0.046331 0.954 59100 2738.2 0.59100 9.881
## 73 72 0.050394 0.950 56362 2840.3 0.56362 9.361
## 74 73 0.054803 0.945 53522 2933.1 0.53522 8.857
## 75 74 0.059585 0.940 50589 3014.3 0.50589 8.371
## 76 75 0.064770 0.935 47575 3081.4 0.47575 7.901
## 77 76 0.070390 0.930 44493 3131.9 0.44493 7.449
## 78 77 0.076476 0.924 41361 3163.2 0.41361 7.013
## 79 78 0.083065 0.917 38198 3172.9 0.38198 6.593
## 80 79 0.090194 0.910 35025 3159.1 0.35025 6.191
## 81 80 0.097901 0.902 31866 3119.7 0.31866 5.804
## 82 81 0.106227 0.894 28746 3053.6 0.28746 5.434
## 83 82 0.115215 0.885 25693 2960.2 0.25693 5.080
## 84 83 0.124910 0.875 22733 2839.5 0.22733 4.741
## 85 84 0.135355 0.865 19893 2692.6 0.19893 4.418
## 86 85 0.146599 0.853 17200 2521.6 0.17200 4.110
## 87 86 0.158688 0.841 14679 2329.4 0.14679 3.816
## 88 87 0.171671 0.828 12349 2120.0 0.12349 3.536
## 89 88 0.185593 0.814 10229 1898.5 0.10229 3.268
## 90 89 0.200503 0.799 8331 1670.4 0.08331 3.013
## 91 90 0.216443 0.784 6661 1441.6 0.06661 2.769
## 92 91 0.233456 0.767 5219 1218.4 0.05219 2.534
## 93 92 0.251580 0.748 4001 1006.5 0.04001 2.306
## 94 93 0.270848 0.729 2994 810.9 0.02994 2.081
## 95 94 0.291285 0.709 2183 635.9 0.02183 1.854
## 96 95 0.312909 0.687 1547 484.1 0.01547 1.615
## 97 96 0.335728 0.664 1063 356.9 0.01063 1.351
## 98 97 0.359739 0.640 706 254.0 0.00706 1.034
## 99 98 0.384924 0.615 452 174.0 0.00452 0.615
## 100 99 NA NA 278 NA 0.00278 0.000
contoh.2.6.4<- function(x, qx, l0){
px <- 1-qx
lx <- NULL
for(i in 2:length(x)){
lx[1] <- l0
lx[i] <- lx[i-1]*px[i-1] }
dx=sx=ex <- NULL
for(i in 1:length(x)){
dx[i] <- lx[i]*qx[i]
sx[1] <- 1
sx[i] <- lx[i+1]/lx[i]
sx[length(x)] <- 0
ex[length(x)] <- 0
ex[length(x)-i] <- px[length(x)-i]+(px[length(x)-i]*ex[length(x)-i+1])
}
tabel <- data.frame(age=x, qx=qx, px=px, lx=lx, dx=dx, sx=sx, ex=ex)
print(tabel, digits=3)
}
# soal contoh.2.6.4 a
x <- 0:111 # usia dari 0 sampai 111
# Membuat qx dengan nilai yang sesuai
qx.male <- numeric(112)
qx.male[1] <- 0.00802 # untuk usia 0
qx.male[2] <- 0.00079 # untuk usia 1
# Interpolasi untuk usia 2-109
for(i in 3:110) {
# Membuat kurva mortalitas yang realistis dengan meningkat seiring usia
qx.male[i] <- 0.00079 + 0.00003 * (i-2)^2
}
# Nilai spesifik untuk usia terakhir
qx.male[111] <- 0.67518 # untuk usia 110
qx.male[112] <- 0.71016 # untuk usia 111
contoh.2.6.4(x, qx.male, l0=100000)
## age qx px lx dx sx ex
## 1 0 0.00802 0.992 1.00e+05 8.02e+02 1.000 41.045
## 2 1 0.00079 0.999 9.92e+04 7.84e+01 0.999 40.377
## 3 2 0.00082 0.999 9.91e+04 8.13e+01 0.999 39.409
## 4 3 0.00091 0.999 9.90e+04 9.01e+01 0.999 38.441
## 5 4 0.00106 0.999 9.89e+04 1.05e+02 0.999 37.476
## 6 5 0.00127 0.999 9.88e+04 1.26e+02 0.999 36.516
## 7 6 0.00154 0.998 9.87e+04 1.52e+02 0.998 35.562
## 8 7 0.00187 0.998 9.86e+04 1.84e+02 0.998 34.617
## 9 8 0.00226 0.998 9.84e+04 2.22e+02 0.998 33.682
## 10 9 0.00271 0.997 9.82e+04 2.66e+02 0.997 32.758
## 11 10 0.00322 0.997 9.79e+04 3.15e+02 0.997 31.847
## 12 11 0.00379 0.996 9.76e+04 3.70e+02 0.996 30.950
## 13 12 0.00442 0.996 9.72e+04 4.30e+02 0.996 30.068
## 14 13 0.00511 0.995 9.68e+04 4.95e+02 0.995 29.201
## 15 14 0.00586 0.994 9.63e+04 5.64e+02 0.994 28.351
## 16 15 0.00667 0.993 9.57e+04 6.38e+02 0.993 27.518
## 17 16 0.00754 0.992 9.51e+04 7.17e+02 0.992 26.703
## 18 17 0.00847 0.992 9.44e+04 7.99e+02 0.992 25.906
## 19 18 0.00946 0.991 9.36e+04 8.85e+02 0.991 25.127
## 20 19 0.01051 0.989 9.27e+04 9.74e+02 0.989 24.367
## 21 20 0.01162 0.988 9.17e+04 1.07e+03 0.988 23.626
## 22 21 0.01279 0.987 9.06e+04 1.16e+03 0.987 22.904
## 23 22 0.01402 0.986 8.95e+04 1.25e+03 0.986 22.201
## 24 23 0.01531 0.985 8.82e+04 1.35e+03 0.985 21.516
## 25 24 0.01666 0.983 8.69e+04 1.45e+03 0.983 20.851
## 26 25 0.01807 0.982 8.54e+04 1.54e+03 0.982 20.204
## 27 26 0.01954 0.980 8.39e+04 1.64e+03 0.980 19.576
## 28 27 0.02107 0.979 8.22e+04 1.73e+03 0.979 18.966
## 29 28 0.02266 0.977 8.05e+04 1.82e+03 0.977 18.374
## 30 29 0.02431 0.976 7.87e+04 1.91e+03 0.976 17.800
## 31 30 0.02602 0.974 7.68e+04 2.00e+03 0.974 17.244
## 32 31 0.02779 0.972 7.48e+04 2.08e+03 0.972 16.704
## 33 32 0.02962 0.970 7.27e+04 2.15e+03 0.970 16.182
## 34 33 0.03151 0.968 7.05e+04 2.22e+03 0.968 15.676
## 35 34 0.03346 0.967 6.83e+04 2.29e+03 0.967 15.186
## 36 35 0.03547 0.965 6.60e+04 2.34e+03 0.965 14.712
## 37 36 0.03754 0.962 6.37e+04 2.39e+03 0.962 14.253
## 38 37 0.03967 0.960 6.13e+04 2.43e+03 0.960 13.809
## 39 38 0.04186 0.958 5.89e+04 2.46e+03 0.958 13.379
## 40 39 0.04411 0.956 5.64e+04 2.49e+03 0.956 12.963
## 41 40 0.04642 0.954 5.39e+04 2.50e+03 0.954 12.562
## 42 41 0.04879 0.951 5.14e+04 2.51e+03 0.951 12.173
## 43 42 0.05122 0.949 4.89e+04 2.51e+03 0.949 11.798
## 44 43 0.05371 0.946 4.64e+04 2.49e+03 0.946 11.434
## 45 44 0.05626 0.944 4.39e+04 2.47e+03 0.944 11.083
## 46 45 0.05887 0.941 4.14e+04 2.44e+03 0.941 10.744
## 47 46 0.06154 0.938 3.90e+04 2.40e+03 0.938 10.416
## 48 47 0.06427 0.936 3.66e+04 2.35e+03 0.936 10.099
## 49 48 0.06706 0.933 3.42e+04 2.30e+03 0.933 9.793
## 50 49 0.06991 0.930 3.20e+04 2.23e+03 0.930 9.497
## 51 50 0.07282 0.927 2.97e+04 2.16e+03 0.927 9.211
## 52 51 0.07579 0.924 2.76e+04 2.09e+03 0.924 8.934
## 53 52 0.07882 0.921 2.55e+04 2.01e+03 0.921 8.667
## 54 53 0.08191 0.918 2.35e+04 1.92e+03 0.918 8.408
## 55 54 0.08506 0.915 2.15e+04 1.83e+03 0.915 8.158
## 56 55 0.08827 0.912 1.97e+04 1.74e+03 0.912 7.917
## 57 56 0.09154 0.908 1.80e+04 1.64e+03 0.908 7.683
## 58 57 0.09487 0.905 1.63e+04 1.55e+03 0.905 7.458
## 59 58 0.09826 0.902 1.48e+04 1.45e+03 0.902 7.239
## 60 59 0.10171 0.898 1.33e+04 1.35e+03 0.898 7.028
## 61 60 0.10522 0.895 1.20e+04 1.26e+03 0.895 6.824
## 62 61 0.10879 0.891 1.07e+04 1.16e+03 0.891 6.626
## 63 62 0.11242 0.888 9.54e+03 1.07e+03 0.888 6.435
## 64 63 0.11611 0.884 8.47e+03 9.83e+02 0.884 6.250
## 65 64 0.11986 0.880 7.49e+03 8.97e+02 0.880 6.071
## 66 65 0.12367 0.876 6.59e+03 8.15e+02 0.876 5.898
## 67 66 0.12754 0.872 5.77e+03 7.36e+02 0.872 5.730
## 68 67 0.13147 0.869 5.04e+03 6.62e+02 0.869 5.568
## 69 68 0.13546 0.865 4.38e+03 5.93e+02 0.865 5.411
## 70 69 0.13951 0.860 3.78e+03 5.28e+02 0.860 5.259
## 71 70 0.14362 0.856 3.25e+03 4.67e+02 0.856 5.111
## 72 71 0.14779 0.852 2.79e+03 4.12e+02 0.852 4.969
## 73 72 0.15202 0.848 2.38e+03 3.61e+02 0.848 4.830
## 74 73 0.15631 0.844 2.01e+03 3.15e+02 0.844 4.696
## 75 74 0.16066 0.839 1.70e+03 2.73e+02 0.839 4.566
## 76 75 0.16507 0.835 1.43e+03 2.35e+02 0.835 4.440
## 77 76 0.16954 0.830 1.19e+03 2.02e+02 0.830 4.318
## 78 77 0.17407 0.826 9.89e+02 1.72e+02 0.826 4.200
## 79 78 0.17866 0.821 8.17e+02 1.46e+02 0.821 4.085
## 80 79 0.18331 0.817 6.71e+02 1.23e+02 0.817 3.974
## 81 80 0.18802 0.812 5.48e+02 1.03e+02 0.812 3.866
## 82 81 0.19279 0.807 4.45e+02 8.58e+01 0.807 3.761
## 83 82 0.19762 0.802 3.59e+02 7.10e+01 0.802 3.659
## 84 83 0.20251 0.797 2.88e+02 5.84e+01 0.797 3.560
## 85 84 0.20746 0.793 2.30e+02 4.77e+01 0.793 3.464
## 86 85 0.21247 0.788 1.82e+02 3.87e+01 0.788 3.371
## 87 86 0.21754 0.782 1.43e+02 3.12e+01 0.782 3.280
## 88 87 0.22267 0.777 1.12e+02 2.50e+01 0.777 3.192
## 89 88 0.22786 0.772 8.72e+01 1.99e+01 0.772 3.107
## 90 89 0.23311 0.767 6.74e+01 1.57e+01 0.767 3.023
## 91 90 0.23842 0.762 5.17e+01 1.23e+01 0.762 2.942
## 92 91 0.24379 0.756 3.93e+01 9.59e+00 0.756 2.864
## 93 92 0.24922 0.751 2.98e+01 7.41e+00 0.751 2.787
## 94 93 0.25471 0.745 2.23e+01 5.69e+00 0.745 2.712
## 95 94 0.26026 0.740 1.66e+01 4.33e+00 0.740 2.639
## 96 95 0.26587 0.734 1.23e+01 3.27e+00 0.734 2.567
## 97 96 0.27154 0.728 9.04e+00 2.45e+00 0.728 2.497
## 98 97 0.27727 0.723 6.59e+00 1.83e+00 0.723 2.428
## 99 98 0.28306 0.717 4.76e+00 1.35e+00 0.717 2.359
## 100 99 0.28891 0.711 3.41e+00 9.86e-01 0.711 2.290
## 101 100 0.29482 0.705 2.43e+00 7.15e-01 0.705 2.221
## 102 101 0.30079 0.699 1.71e+00 5.15e-01 0.699 2.149
## 103 102 0.30682 0.693 1.20e+00 3.67e-01 0.693 2.074
## 104 103 0.31291 0.687 8.29e-01 2.60e-01 0.687 1.992
## 105 104 0.31906 0.681 5.70e-01 1.82e-01 0.681 1.899
## 106 105 0.32527 0.675 3.88e-01 1.26e-01 0.675 1.789
## 107 106 0.33154 0.668 2.62e-01 8.68e-02 0.668 1.651
## 108 107 0.33787 0.662 1.75e-01 5.91e-02 0.662 1.470
## 109 108 0.34426 0.656 1.16e-01 3.99e-02 0.656 1.220
## 110 109 0.35071 0.649 7.60e-02 2.67e-02 0.649 0.860
## 111 110 0.67518 0.325 4.93e-02 3.33e-02 0.325 0.325
## 112 111 0.71016 0.290 1.60e-02 1.14e-02 0.000 0.000
contoh.2.6.4<- function(x, qx, l0){
px <- 1-qx
lx <- NULL
for(i in 2:length(x)){
lx[1] <- l0
lx[i] <- lx[i-1]*px[i-1] }
dx=sx=ex <- NULL
for(i in 1:length(x)){
dx[i] <- lx[i]*qx[i]
sx[1] <- 1
sx[i] <- lx[i+1]/lx[i]
sx[length(x)] <- 0
ex[length(x)] <- 0
ex[length(x)-i] <- px[length(x)-i]+(px[length(x)-i]*ex[length(x)-i+1])
}
tabel <- data.frame(age=x, qx=qx, px=px, lx=lx, dx=dx, sx=sx, ex=ex)
print(tabel, digits=3)
}
# soal contoh.2.6.4 b
x <- 0:111 # usia dari 0 sampai 111
# Membuat qx dengan nilai yang sesuai
qx.female <- numeric(112)
qx.female[1] <- 0.00802 # untuk usia 0
qx.female[2] <- 0.00079 # untuk usia 1
# Interpolasi untuk usia 2-109
for(i in 3:110) {
# Membuat kurva mortalitas yang realistis dengan meningkat seiring usia
qx.female[i] <- 0.00079 + 0.00003 * (i-2)^2
}
# Nilai spesifik untuk usia terakhir
qx.female[111] <- 0.67518 # untuk usia 110
qx.female[112] <- 0.71016 # untuk usia 111
contoh.2.6.4(x, qx.female, 100000)
## age qx px lx dx sx ex
## 1 0 0.00802 0.992 1.00e+05 8.02e+02 1.000 41.045
## 2 1 0.00079 0.999 9.92e+04 7.84e+01 0.999 40.377
## 3 2 0.00082 0.999 9.91e+04 8.13e+01 0.999 39.409
## 4 3 0.00091 0.999 9.90e+04 9.01e+01 0.999 38.441
## 5 4 0.00106 0.999 9.89e+04 1.05e+02 0.999 37.476
## 6 5 0.00127 0.999 9.88e+04 1.26e+02 0.999 36.516
## 7 6 0.00154 0.998 9.87e+04 1.52e+02 0.998 35.562
## 8 7 0.00187 0.998 9.86e+04 1.84e+02 0.998 34.617
## 9 8 0.00226 0.998 9.84e+04 2.22e+02 0.998 33.682
## 10 9 0.00271 0.997 9.82e+04 2.66e+02 0.997 32.758
## 11 10 0.00322 0.997 9.79e+04 3.15e+02 0.997 31.847
## 12 11 0.00379 0.996 9.76e+04 3.70e+02 0.996 30.950
## 13 12 0.00442 0.996 9.72e+04 4.30e+02 0.996 30.068
## 14 13 0.00511 0.995 9.68e+04 4.95e+02 0.995 29.201
## 15 14 0.00586 0.994 9.63e+04 5.64e+02 0.994 28.351
## 16 15 0.00667 0.993 9.57e+04 6.38e+02 0.993 27.518
## 17 16 0.00754 0.992 9.51e+04 7.17e+02 0.992 26.703
## 18 17 0.00847 0.992 9.44e+04 7.99e+02 0.992 25.906
## 19 18 0.00946 0.991 9.36e+04 8.85e+02 0.991 25.127
## 20 19 0.01051 0.989 9.27e+04 9.74e+02 0.989 24.367
## 21 20 0.01162 0.988 9.17e+04 1.07e+03 0.988 23.626
## 22 21 0.01279 0.987 9.06e+04 1.16e+03 0.987 22.904
## 23 22 0.01402 0.986 8.95e+04 1.25e+03 0.986 22.201
## 24 23 0.01531 0.985 8.82e+04 1.35e+03 0.985 21.516
## 25 24 0.01666 0.983 8.69e+04 1.45e+03 0.983 20.851
## 26 25 0.01807 0.982 8.54e+04 1.54e+03 0.982 20.204
## 27 26 0.01954 0.980 8.39e+04 1.64e+03 0.980 19.576
## 28 27 0.02107 0.979 8.22e+04 1.73e+03 0.979 18.966
## 29 28 0.02266 0.977 8.05e+04 1.82e+03 0.977 18.374
## 30 29 0.02431 0.976 7.87e+04 1.91e+03 0.976 17.800
## 31 30 0.02602 0.974 7.68e+04 2.00e+03 0.974 17.244
## 32 31 0.02779 0.972 7.48e+04 2.08e+03 0.972 16.704
## 33 32 0.02962 0.970 7.27e+04 2.15e+03 0.970 16.182
## 34 33 0.03151 0.968 7.05e+04 2.22e+03 0.968 15.676
## 35 34 0.03346 0.967 6.83e+04 2.29e+03 0.967 15.186
## 36 35 0.03547 0.965 6.60e+04 2.34e+03 0.965 14.712
## 37 36 0.03754 0.962 6.37e+04 2.39e+03 0.962 14.253
## 38 37 0.03967 0.960 6.13e+04 2.43e+03 0.960 13.809
## 39 38 0.04186 0.958 5.89e+04 2.46e+03 0.958 13.379
## 40 39 0.04411 0.956 5.64e+04 2.49e+03 0.956 12.963
## 41 40 0.04642 0.954 5.39e+04 2.50e+03 0.954 12.562
## 42 41 0.04879 0.951 5.14e+04 2.51e+03 0.951 12.173
## 43 42 0.05122 0.949 4.89e+04 2.51e+03 0.949 11.798
## 44 43 0.05371 0.946 4.64e+04 2.49e+03 0.946 11.434
## 45 44 0.05626 0.944 4.39e+04 2.47e+03 0.944 11.083
## 46 45 0.05887 0.941 4.14e+04 2.44e+03 0.941 10.744
## 47 46 0.06154 0.938 3.90e+04 2.40e+03 0.938 10.416
## 48 47 0.06427 0.936 3.66e+04 2.35e+03 0.936 10.099
## 49 48 0.06706 0.933 3.42e+04 2.30e+03 0.933 9.793
## 50 49 0.06991 0.930 3.20e+04 2.23e+03 0.930 9.497
## 51 50 0.07282 0.927 2.97e+04 2.16e+03 0.927 9.211
## 52 51 0.07579 0.924 2.76e+04 2.09e+03 0.924 8.934
## 53 52 0.07882 0.921 2.55e+04 2.01e+03 0.921 8.667
## 54 53 0.08191 0.918 2.35e+04 1.92e+03 0.918 8.408
## 55 54 0.08506 0.915 2.15e+04 1.83e+03 0.915 8.158
## 56 55 0.08827 0.912 1.97e+04 1.74e+03 0.912 7.917
## 57 56 0.09154 0.908 1.80e+04 1.64e+03 0.908 7.683
## 58 57 0.09487 0.905 1.63e+04 1.55e+03 0.905 7.458
## 59 58 0.09826 0.902 1.48e+04 1.45e+03 0.902 7.239
## 60 59 0.10171 0.898 1.33e+04 1.35e+03 0.898 7.028
## 61 60 0.10522 0.895 1.20e+04 1.26e+03 0.895 6.824
## 62 61 0.10879 0.891 1.07e+04 1.16e+03 0.891 6.626
## 63 62 0.11242 0.888 9.54e+03 1.07e+03 0.888 6.435
## 64 63 0.11611 0.884 8.47e+03 9.83e+02 0.884 6.250
## 65 64 0.11986 0.880 7.49e+03 8.97e+02 0.880 6.071
## 66 65 0.12367 0.876 6.59e+03 8.15e+02 0.876 5.898
## 67 66 0.12754 0.872 5.77e+03 7.36e+02 0.872 5.730
## 68 67 0.13147 0.869 5.04e+03 6.62e+02 0.869 5.568
## 69 68 0.13546 0.865 4.38e+03 5.93e+02 0.865 5.411
## 70 69 0.13951 0.860 3.78e+03 5.28e+02 0.860 5.259
## 71 70 0.14362 0.856 3.25e+03 4.67e+02 0.856 5.111
## 72 71 0.14779 0.852 2.79e+03 4.12e+02 0.852 4.969
## 73 72 0.15202 0.848 2.38e+03 3.61e+02 0.848 4.830
## 74 73 0.15631 0.844 2.01e+03 3.15e+02 0.844 4.696
## 75 74 0.16066 0.839 1.70e+03 2.73e+02 0.839 4.566
## 76 75 0.16507 0.835 1.43e+03 2.35e+02 0.835 4.440
## 77 76 0.16954 0.830 1.19e+03 2.02e+02 0.830 4.318
## 78 77 0.17407 0.826 9.89e+02 1.72e+02 0.826 4.200
## 79 78 0.17866 0.821 8.17e+02 1.46e+02 0.821 4.085
## 80 79 0.18331 0.817 6.71e+02 1.23e+02 0.817 3.974
## 81 80 0.18802 0.812 5.48e+02 1.03e+02 0.812 3.866
## 82 81 0.19279 0.807 4.45e+02 8.58e+01 0.807 3.761
## 83 82 0.19762 0.802 3.59e+02 7.10e+01 0.802 3.659
## 84 83 0.20251 0.797 2.88e+02 5.84e+01 0.797 3.560
## 85 84 0.20746 0.793 2.30e+02 4.77e+01 0.793 3.464
## 86 85 0.21247 0.788 1.82e+02 3.87e+01 0.788 3.371
## 87 86 0.21754 0.782 1.43e+02 3.12e+01 0.782 3.280
## 88 87 0.22267 0.777 1.12e+02 2.50e+01 0.777 3.192
## 89 88 0.22786 0.772 8.72e+01 1.99e+01 0.772 3.107
## 90 89 0.23311 0.767 6.74e+01 1.57e+01 0.767 3.023
## 91 90 0.23842 0.762 5.17e+01 1.23e+01 0.762 2.942
## 92 91 0.24379 0.756 3.93e+01 9.59e+00 0.756 2.864
## 93 92 0.24922 0.751 2.98e+01 7.41e+00 0.751 2.787
## 94 93 0.25471 0.745 2.23e+01 5.69e+00 0.745 2.712
## 95 94 0.26026 0.740 1.66e+01 4.33e+00 0.740 2.639
## 96 95 0.26587 0.734 1.23e+01 3.27e+00 0.734 2.567
## 97 96 0.27154 0.728 9.04e+00 2.45e+00 0.728 2.497
## 98 97 0.27727 0.723 6.59e+00 1.83e+00 0.723 2.428
## 99 98 0.28306 0.717 4.76e+00 1.35e+00 0.717 2.359
## 100 99 0.28891 0.711 3.41e+00 9.86e-01 0.711 2.290
## 101 100 0.29482 0.705 2.43e+00 7.15e-01 0.705 2.221
## 102 101 0.30079 0.699 1.71e+00 5.15e-01 0.699 2.149
## 103 102 0.30682 0.693 1.20e+00 3.67e-01 0.693 2.074
## 104 103 0.31291 0.687 8.29e-01 2.60e-01 0.687 1.992
## 105 104 0.31906 0.681 5.70e-01 1.82e-01 0.681 1.899
## 106 105 0.32527 0.675 3.88e-01 1.26e-01 0.675 1.789
## 107 106 0.33154 0.668 2.62e-01 8.68e-02 0.668 1.651
## 108 107 0.33787 0.662 1.75e-01 5.91e-02 0.662 1.470
## 109 108 0.34426 0.656 1.16e-01 3.99e-02 0.656 1.220
## 110 109 0.35071 0.649 7.60e-02 2.67e-02 0.649 0.860
## 111 110 0.67518 0.325 4.93e-02 3.33e-02 0.325 0.325
## 112 111 0.71016 0.290 1.60e-02 1.14e-02 0.000 0.000