Let X be standard normally distributed, i.e., X N (0, 1). The first problem prov
ID: 3074306 • Letter: L
Question
Let X be standard normally distributed, i.e.,
X N (0, 1). The first problem provides you with a formula for
E[(e^XK)+], K0. (1)
For K {0.1, 0.5, 1, 2, 5} you are then asked to compute the expectation (1) using Monte Carlo simulation. Generate a sample of X using, e.g., Excel by proceeding as follows: First generate a random sample of a uniform in Excel using the command RAND and then transform this number into a N(0,1)-number. This is done by applying the NORMINV command in Excel. Please note that NORMINV takes standard deviation and not variance as input. You need to generate a lot of samples which we call xn, n = 1,...,N. You can then use
1/N(e^xn K)+,
as an estimate of the expectation appearing in (1).
Explanation / Answer
IF you start with a seed value, you will not get a changing number everytime
I have taken about a 1000 simulations but even those show some amount of variability in the average distance.
However, one can safely say from the results of mean distances, that the required value is about 0.5.
Note : The table that I have simulated is shown here. It only has 200 simulations because chegg editor does not allow to let me have answers over 65,000 characters.
Ask if you have any doubts.
X1 Y1 X2 Y2 Distance 0.0111 0.0371 0.1653 0.5098 0.4972 Average 0.7478 0.3435 0.7537 0.0519 0.2916 0.5063 0.7083 0.1718 0.7273 0.9920 0.8204 0.5974 0.5040 0.3518 0.3975 0.2676 0.1150 0.0315 0.6784 0.2396 0.6006 0.4775 0.2173 0.4825 0.1975 0.0205 0.1280 0.7891 0.9099 0.3482 0.8977 0.7929 0.1953 0.2328 0.5222 0.6485 0.5398 0.0557 0.6239 0.6063 0.5570 0.9942 0.1368 0.3542 0.5748 0.7756 0.3704 0.8844 0.3212 0.5307 0.3571 0.4727 0.0438 0.3129 0.5382 0.5196 0.3245 0.3992 0.9150 0.5265 0.6041 0.5796 0.1632 0.4661 0.0482 0.1616 0.3737 0.0488 0.9691 0.1720 0.6080 0.7710 0.5530 0.3624 0.2911 0.4853 0.4632 0.4859 0.0607 0.1295 0.5376 0.8767 0.2738 0.1676 0.7262 0.8411 0.3440 0.0706 0.2696 0.6798 0.6137 0.4957 0.2052 0.2158 0.4335 0.3611 0.5299 0.9998 0.7906 0.8827 0.2858 0.5148 0.4871 0.6849 0.0863 0.4354 0.3797 0.0605 0.6723 0.3622 0.4203 0.7537 0.5862 0.3973 0.5492 0.3583 0.3040 0.0095 0.6801 0.7163 0.8007 0.7344 0.9873 0.7461 0.1002 0.8872 0.0971 0.5764 0.2847 0.5527 0.1891 0.5756 0.5100 0.7454 0.2789 0.2867 0.3946 0.0934 0.7752 0.1484 0.3846 0.0125 0.2498 0.2063 0.3588 0.2223 0.2548 0.7484 0.4976 0.7353 0.2432 0.5411 0.1446 0.0380 0.4450 0.5860 0.8661 0.4646 0.2177 0.4099 0.6508 0.6876 0.1660 0.6728 0.1970 0.0344 0.3790 0.4040 0.6099 0.6691 0.3516 0.4033 0.1361 0.6176 0.1763 0.2180 0.8631 0.0962 0.8700 0.5698 0.4737 0.5841 0.1684 0.1294 0.8191 0.7938 0.3887 0.1928 0.4538 0.5141 0.3279 0.7146 0.7344 0.0161 0.0335 0.9895 0.2349 0.8458 0.9403 0.6598 0.7295 0.6380 0.3552 0.5622 0.4232 0.1018 0.2203 0.7118 0.0198 0.9553 0.3154 0.4507 0.9178 0.5798 0.1338 0.7945 0.1612 0.5808 0.6074 0.7996 0.4970 0.5216 0.6161 0.5842 0.3267 0.2960 0.4528 0.6302 0.0814 0.3280 0.4788 0.6366 0.5180 0.0946 0.2760 0.5936 0.7767 0.3035 0.6460 0.3217 0.1319 0.2684 0.7563 0.0602 0.0913 0.6968 0.9327 0.8145 0.8203 0.7433 0.1330 0.9709 0.3454 0.4106 0.0656 0.6262 0.7961 0.2759 0.0512 0.8966 0.9696 0.8985 0.6225 0.7321 0.6477 0.1683 0.3992 0.4534 0.4517 0.2123 0.2468 0.4510 0.3139 0.0677 0.0398 0.4712 0.1392 0.1447 0.6642 0.6576 0.7339 0.0429 0.2269 0.8437 0.4206 0.8238 0.1092 0.1830 0.8487 0.7717 0.9452 0.9983 0.5658 0.9199 0.7924 0.2398 0.5560 0.0665 0.7964 0.3259 0.3537 0.5695 0.8733 0.7657 0.4540 0.4629 0.4618 0.2116 0.1676 0.6086 0.4941 0.0828 0.5970 0.7608 0.0986 0.8415 0.2514 0.8325 0.8554 0.3356 0.7821 0.2070 0.2810 0.6209 0.5361 0.4862 0.6241 0.4447 0.4733 0.1853 0.3000 0.3485 0.3819 0.9485 0.0408 0.6902 0.5106 0.9939 0.9691 0.9588 0.4599 0.1707 0.7107 0.9089 0.3241 0.8333 0.1554 0.6100 0.3632 0.3648 0.3215 0.2383 0.4985 0.0306 0.1324 0.4209 0.3236 0.9863 0.6115 0.4303 0.6262 0.7872 0.0887 0.1314 0.4694 0.7583 0.6328 0.3828 0.5937 0.5352 0.1574 0.6187 0.2530 0.3690 0.4440 0.3143 0.0106 0.2555 0.2269 0.0729 0.2831 0.1193 0.1328 0.4169 0.7426 0.6786 0.2760 0.5103 0.7038 0.7351 0.4832 0.4051 0.8056 0.7651 0.7274 0.3684 0.8470 0.3818 0.8556 0.4223 0.0414 0.5913 0.4765 0.1459 0.1583 0.5474 0.5723 0.9372 0.6409 0.7786 0.1728 0.1868 0.2946 0.0847 0.5412 0.2670 0.6882 0.2052 0.7584 0.3520 0.1627 0.6137 0.7943 0.5759 0.1003 0.6950 0.9012 0.9302 0.3857 0.3799 0.7541 0.8519 0.9620 0.8834 0.3758 0.5870 0.5946 0.9507 0.0420 0.1986 0.9333 0.2085 0.9269 0.4664 0.4467 0.5451 0.1603 0.2036 0.5770 0.2789 0.4234 0.2123 0.2782 0.0306 0.7781 0.5319 0.4567 0.3530 0.2837 0.8863 0.5607 0.6338 0.0059 0.7037 0.6944 0.6921 0.3896 0.0438 0.1368 0.3659 0.4094 0.6289 0.0674 0.3359 0.4766 0.5032 0.6201 0.7814 0.1564 0.0524 0.8640 0.0091 0.5790 0.9262 0.0848 1.0418 0.3964 0.4345 0.8955 0.8062 0.6222 0.6360 0.6321 0.1728 0.5064 0.4800 0.5106 0.3650 0.0539 0.1721 0.4958 0.9404 0.7348 0.1491 0.7283 0.7913 0.3810 0.1815 0.2163 0.1658 0.1655 0.5672 0.8782 0.1044 0.3345 0.7140 0.2589 0.1180 0.7156 0.6833 0.7267 0.1448 0.2279 0.3898 0.7293 0.5580 0.5610 0.0381 0.6210 0.5362 0.5017 0.1797 0.4403 0.5085 0.9843 0.6356 0.3259 0.0639 0.8515 0.1119 0.5278 0.0759 0.6041 0.2923 0.3114 0.3639 0.7966 0.8298 0.4143 0.6590 0.4188 0.2795 0.8065 0.2019 0.1608 0.6504 0.1274 0.3613 0.7676 0.1968 0.6610 0.5484 0.3249 0.7529 0.9427 0.6507 0.2157 0.5793 0.3093 0.0513 0.5362 0.3466 0.4433 0.0662 0.3759 0.2884 0.6595 0.4284 0.9903 0.0146 0.5297 0.3637 0.8933 0.5445 0.4599 0.4697 0.8388 0.6315 0.0523 0.1307 0.9324 0.1596 0.9210 0.6724 0.0728 0.9911 0.1232 0.0682 0.8918 0.1050 0.7695 0.9341 0.9779 0.9425 0.1941 0.7839 0.2038 0.7788 0.0478 0.3401 0.4657 0.4211 0.0379 0.9059 0.7518 0.8629 0.5498 0.5935 0.3152 0.5954 0.2346 0.9085 0.5384 0.3486 0.0956 0.7139 0.9912 0.3858 0.9833 0.2727 0.1134 0.3825 0.9465 0.5915 0.0944 0.8773 0.4254 0.9518 0.1989 0.3261 0.6654 0.7971 0.9507 0.9169 0.8588 0.1509 0.6493 0.5141 0.4547 0.7973 0.3436 0.4689 0.2771 0.1185 0.0889 0.3978 0.3164 0.8504 0.2024 0.9755 0.1692 0.5964 0.1313 0.1782 0.0471 0.4266 0.0188 0.0478 0.5911 0.1828 0.5881 0.9045 0.2641 0.0658 0.1554 0.8458 0.4942 0.8814 0.3306 0.1845 0.7158 0.4142 0.6633 0.6655 0.2182 0.5112 0.9579 0.8184 0.6798 0.2239 0.6564 0.3370 0.8701 0.8265 0.2703 0.7742 0.5601 0.6467 0.5594 0.6824 0.0357 0.0945 0.9204 0.5386 0.0489 0.9781 0.9676 0.7293 0.7911 0.3652 0.4046 0.4631 0.9113 0.2333 0.0919 0.8510 0.0709 0.8828 0.4601 0.4110 0.6116 0.8431 0.1924 0.2092 0.4601 0.6881 0.8163 0.8814 0.4039 0.8272 0.4160 0.6534 0.1823 0.1832 0.0851 0.4801 0.1630 0.5861 0.8250 0.6501 0.6651 0.8515 0.8101 0.9661 0.7419 0.1333 0.1650 0.6980 0.0298 0.7303 0.1390 0.5772 0.5815 0.2631 0.6465 0.3208 0.1783 0.1377 0.2917 0.1425 0.1135 0.9230 0.2705 0.5999 0.6785 0.5205 0.1925 0.0492 0.8391 0.2207 0.6690 0.8277 0.2355 0.3737 0.2291 0.4541 0.2884 0.4286 0.9690 0.2186 0.7122 0.0969 0.3613 0.0284 0.6733 0.3194 0.2198 0.3779 0.2850 0.5947 0.2264 0.5468 0.6247 0.3557 0.4678 0.2473 0.1204 0.9844 0.0610 0.9222 0.0860 0.2741 0.2865 0.9210 0.8737 0.8736 0.3358 0.8240 0.7291 0.4242 0.5608 0.0504 0.7078 0.6139 0.3039 0.6932 0.3810 0.0905 0.4928 0.1344 0.1201 0.8418 0.7910 0.3242 0.2963 0.7160 0.0218 0.3496 0.0658 0.6814 0.3348 0.4532 0.0469 0.7428 0.1107 0.2965 0.4823 0.9172 0.5349 0.6996 0.2239 0.2983 0.2857 0.2868 0.9520 0.6664 0.7767 0.8641 0.3482 0.8638 0.4285 0.2936 0.9331 0.1511 0.8177 0.1834 0.1664 0.6444 0.4287 0.5499 0.2788 0.8784 0.9725 0.4839 0.9706 0.3945 0.7421 0.3493 0.4124 0.1194 0.4020 0.1590 0.5106 0.6872 0.0364 0.7099 0.1306 0.5867 0.0114 0.8304 0.2712 0.1782 0.9694 0.8431 0.7263 0.7079 0.3270 0.9843 0.5025 0.3717 0.6373 0.3104 0.9022 0.2639 0.1847 0.7190 0.6519 0.3188 0.3406 0.0240 0.4288 0.7802 0.4353 0.1687 0.9621 0.8072 0.8332 0.2701 0.8963 0.4443 0.1852 0.3079 0.3343 0.4834 0.6124 0.3288 0.2129 0.7999 0.3198 0.0765 0.7313 0.8781 0.6953 0.7670 0.9155 0.2467 0.0797 0.1555 0.5292 0.1864 0.4506 0.6352 0.2617 0.2622 0.7256 0.5953 0.1258 0.4990 0.3233 0.5218 0.1989 0.9020 0.3653 0.0712 0.4429 0.8344 0.0285 0.1611 0.6203 0.5976 0.7353 0.3249 0.4928 0.4565 0.1122 0.4027 0.1581 0.5199 0.8209 0.6063 0.6684 0.1433 0.4773 0.2666 0.1450 0.3545 0.4091 0.5049 0.2966 0.9528 0.4617 0.5836 0.3145 0.4869 0.6670 0.3655 0.1575 0.7261 0.0031 0.4146 0.3477 0.3797 0.6044 0.9154 0.4914 0.5475 0.3432 0.2274 0.2185 0.9272 0.7109 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