Taller 3
Angel Cordoba
2025-09-24
Simulación de Variables Aleatorias en R
1. Un sistema de producción tiene fallas según un proceso de Poisson con una tasa de 3 fallas por día.
Simular el número de fallas en un semestre (150 días) y calcular la media y desviación estándar.
#Semilla
set.seed(2804)
#Parametros a utulizar
lambda <-3
n <- 150
#Simulacion de las fallas de cada dia
fallas<- rpois(n,lambda)
fallas ## [1] 3 1 4 1 2 3 1 6 1 11 2 3 4 3 3 3 3 2 1 6 3 3 3 2 3
## [26] 1 5 1 3 4 5 1 2 2 4 2 4 3 4 1 3 2 6 4 5 6 2 2 2 8
## [51] 5 1 1 3 4 2 4 3 4 3 4 3 6 2 5 2 4 2 3 3 2 2 4 4 3
## [76] 2 4 4 3 1 2 2 6 1 5 1 2 4 5 3 1 1 3 3 3 2 5 2 3 3
## [101] 0 3 0 6 5 3 4 4 4 0 4 3 4 9 3 3 2 4 2 0 1 4 4 3 5
## [126] 6 0 3 4 2 2 5 2 2 4 1 3 2 5 3 2 2 5 0 1 3 3 3 3 1#Calcular media y deviacion estandar
media<- mean(fallas)
desviacion<- sd(fallas)
#Resultados de la media y desviacion estandar
cat("Media:",media)## Media: 3.04cat("Desviacion estandar:",desviacion)## Desviacion estandar: 1.721869Interpretacion
Simulando los 150 días, el promedio de fallas por día salió cercano a 3. En otras palabras, la simulación confirma que, en promedio, el sistema presenta unas 3 fallas diarias, con fluctuaciones normales alrededor de ese número.
2. La vida útil (en horas) de un componente electrónico sigue una distribución exponencial con un
promedio de 500 horas. Simular 1000 componentes y estimar la probabilidad de que un componente
dure más de 700 horas.
#Semilla
set.seed(2804)
#Parametros
media<- 500
rate <- 1/500
n<- 1000
#Simulacion de vida util
vida<- rexp(n,rate)
vida## [1] 140.5978447 1187.7294303 311.2542201 1121.5730616 618.9972395
## [6] 29.9360633 1102.3063724 29.4812047 387.0116318 501.3610254
## [11] 83.8158149 2.4749157 495.9329003 980.7495540 216.9491834
## [16] 108.7231354 526.6452413 413.9848837 1333.1626271 74.9612369
## [21] 263.2098943 345.7838895 1110.4747579 247.8141575 916.6970018
## [26] 171.2483468 781.3699673 427.6656483 286.5350812 129.8326383
## [31] 774.8853740 23.6937201 432.3977471 278.3185430 90.4941773
## [36] 250.2781185 566.7629764 556.5925357 332.3090225 390.3240659
## [41] 270.8620119 634.5150815 486.7230057 660.4212918 278.7019440
## [46] 160.9798367 117.2826570 919.5288401 383.9517312 656.6336746
## [51] 43.4612524 1295.4087341 518.6791681 277.3414506 17.5690537
## [56] 792.8125327 535.4919443 787.6417815 13.2779405 353.3387040
## [61] 168.8905179 232.4175446 304.1727562 268.1606109 2995.4672380
## [66] 289.0931356 582.2219713 441.6405300 235.9193764 665.6413087
## [71] 1970.2858803 1627.8156549 650.9037511 2140.7190951 81.8595493
## [76] 192.5195744 359.7815549 623.8340703 85.6125425 301.7171789
## [81] 972.4049978 480.2130264 431.8981525 2810.2665127 120.3828142
## [86] 40.7957705 19.1201095 1114.2659997 187.6084874 129.1996460
## [91] 543.0848147 50.5238089 418.3257563 379.6555693 162.8576828
## [96] 1118.0931245 539.6659788 539.7206316 537.4135626 209.4109140
## [101] 226.9635717 866.9469207 900.8000325 4.3437345 48.4206085
## [106] 328.0459044 962.6502096 422.9523386 165.4557358 22.6298633
## [111] 1832.9223719 884.4259064 268.8314083 68.8930633 1523.2736393
## [116] 453.2431499 718.7136570 324.3817198 121.8617912 174.4653343
## [121] 942.0074690 846.8631776 524.9660336 315.5198593 978.1303324
## [126] 131.5977548 1323.2890544 470.8318151 220.8848502 485.2481922
## [131] 848.1996096 891.4892831 441.3401331 617.1205267 174.6213343
## [136] 622.8408743 120.8798748 1316.5048779 331.4577001 550.0116460
## [141] 172.9067268 480.9935074 319.6144253 627.5809440 109.0201780
## [146] 366.9992297 715.9007834 1374.5185928 2.7321922 98.4030736
## [151] 625.6357734 2049.1343913 60.6117149 524.1079088 157.1801237
## [156] 347.8252888 517.5998761 15.6533390 125.4758330 661.2669546
## [161] 149.4007830 247.6547710 214.9401746 462.5727957 799.7173290
## [166] 142.9470284 257.9392570 734.2039336 437.5269756 292.1172520
## [171] 423.6836517 896.1234605 162.1566119 317.9271694 220.6640245
## [176] 168.2625797 1142.1305658 238.1449770 125.9257502 392.7819282
## [181] 469.9543337 223.8476277 120.3797638 22.7731231 221.2456364
## [186] 267.7512250 2065.5882758 1311.4791074 553.4364851 31.7982487
## [191] 81.6494397 989.6919718 328.2766698 161.9579595 1088.5138962
## [196] 20.8581179 369.8631297 107.6987275 1219.1740507 757.8010857
## [201] 353.3255746 275.4378305 258.3928938 137.1999395 125.1183962
## [206] 648.0353758 59.4686458 61.4823679 969.2003943 435.1175975
## [211] 194.3935070 884.7584864 177.4783018 76.8074554 559.1533091
## [216] 206.5851260 499.2478438 63.6136276 1376.8703052 279.4149786
## [221] 68.2120968 1367.4492603 367.9910611 143.1782944 4.1076199
## [226] 522.7389154 374.0690796 261.0894486 393.3668639 2438.6686318
## [231] 2497.1558259 20.8386504 616.8979457 335.7612547 1162.3369335
## [236] 132.2359528 139.7202068 788.3677211 82.9628527 293.4533022
## [241] 0.4051576 650.6558987 382.4087661 58.3491051 463.3758468
## [246] 49.8204606 165.1219965 119.1650534 750.4301574 444.7677058
## [251] 615.9562589 365.1734666 36.0767767 126.8428746 40.5572858
## [256] 156.5298145 2.8556390 227.0560537 671.2705791 1599.3676371
## [261] 115.4485901 566.0417848 718.8575098 359.9944021 651.4416300
## [266] 243.5037231 142.6223542 63.8889886 23.0038483 52.9184218
## [271] 732.6834034 336.9774907 489.6318561 629.6562259 1522.7193559
## [276] 598.6062949 1479.9630567 606.6023703 512.0507963 1419.5154019
## [281] 639.2200747 241.1614079 72.8144471 169.7526532 770.8184896
## [286] 349.8073366 10.8407964 540.2666768 769.2834819 8.3569377
## [291] 234.0320945 151.6340027 105.4599071 32.8766324 179.6743564
## [296] 574.6570886 633.3007086 2519.4571548 195.7796542 632.5656099
## [301] 807.9742289 515.3600594 1042.3524165 579.5008489 949.3265654
## [306] 435.4050546 376.4996017 236.0885784 282.6312992 451.3902813
## [311] 281.1004175 275.6337967 86.5283071 121.7787827 1418.0181006
## [316] 1089.2017008 76.5527938 244.2487129 1231.3965456 641.9101407
## [321] 676.0490041 740.6005311 35.1233967 2594.2072802 213.4361733
## [326] 6.3568894 295.5704352 529.9124010 905.5530986 49.7008132
## [331] 179.5762444 79.9018775 120.6027521 30.1801478 816.5149130
## [336] 1147.4472386 1463.9151550 129.4377617 243.2728279 359.9425406
## [341] 715.1543049 10.0143351 380.0410950 80.4418502 1099.6579890
## [346] 69.0064752 964.1612153 1196.8255964 484.0098315 80.4031823
## [351] 66.7397056 612.9429559 28.5991934 53.7966623 320.6421649
## [356] 376.4734977 131.7697154 302.8519058 585.9856326 136.3487018
## [361] 542.2959090 177.3478917 493.1034009 564.7729500 124.7785136
## [366] 264.0875676 15.0065999 250.8184437 245.3105072 697.9518635
## [371] 312.2580256 353.1885459 800.6526939 216.9029389 175.2254805
## [376] 1282.5198178 261.0692927 80.4483180 79.6119550 810.3344997
## [381] 165.2393450 139.8599484 145.2561894 55.6030543 19.9766017
## [386] 342.8323958 302.4142189 887.9039911 53.4652062 483.4641669
## [391] 884.1174335 96.7967707 612.8566437 459.6061646 367.1470461
## [396] 59.7136707 9.0912567 121.2453826 472.5044900 843.4746147
## [401] 228.8896814 1523.4924630 280.3815985 548.7680415 530.9955483
## [406] 290.0087570 192.0822576 91.4501588 716.6814271 212.4014227
## [411] 431.8046062 43.1302444 8.9241773 270.5748542 793.4769895
## [416] 363.6215497 115.6167849 614.1240857 1410.5872697 152.1579213
## [421] 862.8902211 661.5487183 727.2440316 106.6097715 327.1160722
## [426] 803.6454171 82.2377350 167.0085806 1284.0246138 995.6254726
## [431] 674.3469904 367.0884557 18.9321238 554.3407262 72.2601737
## [436] 802.1858111 451.4381159 76.1819363 525.1110937 1315.0531323
## [441] 513.4554137 38.6373093 434.1087811 140.3589253 53.0629587
## [446] 89.1022671 828.3604551 252.2917574 189.7186991 91.6472748
## [451] 1069.7163538 2070.9475753 1045.5115726 51.6983689 568.3666747
## [456] 489.6026114 198.3747375 1352.9659909 1311.1806316 977.4161959
## [461] 1393.9658961 16.6849177 366.4219929 119.0221377 471.0159451
## [466] 1255.7223142 1873.5010308 327.4559656 397.1251044 125.7678561
## [471] 51.3734240 185.5576093 691.6314149 499.5633531 150.3538536
## [476] 892.0717490 167.8027082 1005.9145847 64.2275740 320.9163423
## [481] 975.1732312 1052.7077753 1770.8931964 1470.2244837 468.2617334
## [486] 1051.8996728 399.3778163 828.0869907 50.7910121 1027.0575722
## [491] 1413.7093871 118.0040683 622.2813776 964.6961269 479.3454874
## [496] 73.0136779 23.4558890 383.2643031 192.1126249 1063.9751579
## [501] 405.1327547 733.8281786 295.8790392 912.4080147 673.9573059
## [506] 252.3204177 246.4259348 995.1318111 1057.8969633 12.6404858
## [511] 700.5325902 443.9261338 113.3580359 359.5724780 108.2244607
## [516] 280.3000561 64.6102496 387.7284336 85.1266991 182.8980683
## [521] 583.3916329 433.7752922 27.5509641 934.1231091 552.1709658
## [526] 205.6385029 663.7830860 148.2235394 767.0438084 2371.1202535
## [531] 11.2529248 1308.2802286 528.6609679 120.0780824 666.3436242
## [536] 129.6914715 225.9187135 236.7361079 649.5458497 357.7237527
## [541] 133.8673227 511.2606450 106.2994241 172.8026681 360.9698451
## [546] 1532.8977927 270.2627943 520.6970176 181.6157030 415.6800275
## [551] 1052.3045575 683.1503436 335.6639000 456.2084134 1583.9277915
## [556] 534.0979274 300.3933693 1320.0919804 211.8841710 155.7502802
## [561] 458.3063326 766.3037772 511.7039266 334.2570858 26.8986664
## [566] 414.7260664 109.2354139 279.5425234 6.0390106 883.0914851
## [571] 616.9872209 102.0630901 549.3511082 590.6324247 1148.6684521
## [576] 297.3017788 243.3332601 20.1355595 369.2818632 3.5295172
## [581] 379.9416502 600.9240923 442.7820616 48.8306372 300.6436040
## [586] 57.7379678 967.9300692 79.1743153 661.6839203 78.8756127
## [591] 1028.3667091 231.0654882 594.0213199 16.8006702 176.7329315
## [596] 401.2128119 136.8496866 345.2400097 10.2857018 1320.7881646
## [601] 227.3585107 139.7131023 535.8494492 454.2391092 17.9992465
## [606] 467.2009484 858.1848787 30.2217572 299.9940424 611.9032590
## [611] 63.7866685 377.9045935 696.3620865 402.3445109 83.1569371
## [616] 225.1096320 120.7256166 498.1163523 495.4625670 418.4823953
## [621] 333.4201421 57.2725749 609.9850549 620.4724265 2093.2265951
## [626] 152.7763986 633.6070341 2051.3696474 742.8496908 659.2215723
## [631] 1390.0439033 521.6030064 631.0329470 194.8660256 336.0214035
## [636] 538.4011999 1072.2668792 519.4831486 479.9513505 446.8202819
## [641] 497.6115287 251.0416824 343.4446198 477.3174249 656.3851833
## [646] 202.3703535 73.0906522 1174.0397764 68.1858857 254.0622549
## [651] 1108.5651925 696.1101079 561.6106802 618.6451353 79.4121395
## [656] 637.3078692 2062.6554442 393.6866731 1431.3970803 581.9780016
## [661] 2624.8292856 1778.4640472 817.8868644 294.7600142 28.6362139
## [666] 1849.4036448 271.5586261 339.0135318 198.1024090 28.1949483
## [671] 991.3513129 91.8490195 248.8927038 435.0423449 850.3493387
## [676] 469.3445060 385.4813415 432.9900396 362.3782970 59.1838866
## [681] 902.4004712 823.7604778 502.5219396 948.4291058 221.2547415
## [686] 142.3276695 60.9522790 8.1810951 710.6940003 763.8971945
## [691] 1256.3217757 697.2524859 257.8927716 242.0725904 20.3248857
## [696] 857.9959741 820.5995522 670.2060415 60.9428295 57.8218154
## [701] 1057.4988731 85.0952816 603.8883230 233.5666818 651.6847443
## [706] 131.3374753 851.7902391 249.1088964 100.6457230 407.9551831
## [711] 500.8961893 11.5931071 64.4404124 1580.5793702 74.4592114
## [716] 502.6023434 410.6421377 567.8242366 423.0209966 1215.5098263
## [721] 430.3871724 69.0819578 702.9866201 147.7739203 211.6783424
## [726] 79.0994922 12.4669215 275.3071129 196.2767793 934.0186650
## [731] 1120.5556364 179.5299845 959.1395957 12.1200415 183.4256456
## [736] 791.5799953 140.1577877 64.0545788 1306.7763438 1819.0577728
## [741] 12.5603771 249.1679606 1254.9046818 402.2681952 950.9883076
## [746] 735.6823124 1496.8059137 63.8349236 601.8510298 540.5755424
## [751] 330.1305226 397.3945062 373.2472145 179.8452914 1165.2075202
## [756] 448.8078629 431.2190069 435.6682548 521.8233620 999.0670485
## [761] 265.2171878 72.8871170 260.5846863 403.2649169 748.9828020
## [766] 844.5804352 123.3809354 126.8487656 696.4437667 42.1752279
## [771] 975.2985593 498.5417444 264.1162511 543.7701662 39.2990666
## [776] 1453.9043604 85.1260393 571.7663718 354.3503642 287.7982589
## [781] 40.9958458 122.9683820 162.5913759 669.7781854 115.3858495
## [786] 468.3641074 36.5855470 816.6340958 62.6500765 41.5539869
## [791] 6.5317062 16.7895968 301.2333917 1435.4468695 11.7023201
## [796] 191.8920942 673.1069200 1053.3124549 280.7612824 359.4631040
## [801] 1200.3147374 1495.6185929 613.2304310 22.4479297 1443.9807835
## [806] 137.3098564 325.0072801 585.0123274 245.5740923 384.0436116
## [811] 194.2828337 373.2573003 469.5638316 763.9077208 82.1688103
## [816] 156.6565879 1066.1588152 47.1186456 424.5858857 3143.2265427
## [821] 3359.1419771 1963.8424468 387.8274811 63.0453200 563.4325593
## [826] 116.3927896 721.4821921 682.6654468 1044.9734097 272.4767448
## [831] 71.6608525 559.4070004 889.8204928 410.4901375 481.5320833
## [836] 220.9507956 1237.8654191 444.6593367 560.0362213 667.9488318
## [841] 46.8860199 71.1079410 382.5890953 321.5056648 1206.1137901
## [846] 182.3130297 121.2752289 1004.9248375 446.0020464 953.1570002
## [851] 107.9498883 504.3226304 1198.1515353 9.3678508 70.0726607
## [856] 232.5031229 315.0586435 397.5060470 232.2836495 432.0114045
## [861] 304.6068030 262.1230385 1007.4058491 171.7539833 295.7598565
## [866] 729.8778123 342.7802166 22.7921939 12.1683389 503.2351315
## [871] 36.2243871 383.5230293 1982.5185415 17.2896230 228.5793691
## [876] 193.1409542 379.9207401 13.4362297 198.8767469 737.3580141
## [881] 369.1597944 202.3051983 37.2165549 571.3845854 149.9599340
## [886] 572.7037019 322.0022458 614.1265086 341.7368163 136.6540994
## [891] 110.6062899 559.7746963 688.4847749 635.6226420 1209.7125267
## [896] 276.0131375 157.0392726 873.5353094 210.1662045 81.0931816
## [901] 1225.3690851 852.7309733 19.0460951 188.3403566 523.1434321
## [906] 204.5219082 116.8407679 547.5039840 1226.9946364 803.4934345
## [911] 1218.6504238 456.9011479 1280.6510561 324.5179853 371.1595567
## [916] 714.0749978 38.3275375 1267.3450251 1152.1325847 632.3187076
## [921] 151.9070973 2095.1788895 340.6229208 129.0899674 195.7533378
## [926] 143.2698413 2212.5030252 294.6434070 632.7802632 330.0424337
## [931] 384.7679687 613.2273692 150.7207885 703.2162604 168.1004164
## [936] 231.9299640 316.6988583 196.6175617 1676.3289421 870.5910472
## [941] 185.8150985 316.0562909 79.9723174 1.4650785 0.6872180
## [946] 598.3485044 377.5976491 7.7786550 62.7439651 246.6373001
## [951] 101.0105367 578.2812209 440.6502358 171.6629382 251.6983575
## [956] 183.3414868 71.4580768 38.9506306 326.7785558 566.8909172
## [961] 831.3778769 195.2685821 23.3896668 982.7288333 212.5155269
## [966] 655.7513033 103.6264113 516.1368856 219.7991435 443.9463249
## [971] 477.3956570 35.4241197 588.6073064 2106.3412818 590.9288008
## [976] 75.3141032 429.8724639 145.6191768 111.7918838 214.7592050
## [981] 699.6441684 1299.9215358 536.7430355 559.3471308 247.1216170
## [986] 279.3966073 68.0217488 215.7449801 229.1722693 156.5221103
## [991] 593.0326180 377.6334845 67.7617337 520.3927932 48.5319893
## [996] 772.0084143 489.0971131 336.1062822 923.2729170 254.1888468#Probabilidad que dure mas de 700 horas
probabilidad <- mean(vida>700)
#Resultados
cat("Probabilidad que un componente dure mas de 700 horas:",probabilidad)## Probabilidad que un componente dure mas de 700 horas: 0.23Interpretacion
Simulando la vida útil de los 1000 componentes, vimos que solo una parte pequeña supera las 700 horas. Esto tiene sentido porque el promedio es de 500 horas, así que llegar a 700 ya es bastante más arriba de lo esperado.
3. En una línea de ensamblaje, la probabilidad de que un producto sea defectuoso es del 5 %. Simular
100 lotes de 50 productos y calcular el número promedio de productos defectuosos por lote.
#Semilla
set.seed(2804)
#Parametros
n_Producto<- 50
p_defectuoso<- 0.05
n_lotes <- 100
#Simulacion de cantidad de productos defectuosos por lote
defectuosos<- rbinom(n_lotes,n_Producto,p_defectuoso)
defectuosos## [1] 3 1 4 0 2 2 0 5 1 9 1 2 3 2 3 3 2 2 1 5 3 3 3 2 2 1 5 1 3 4 4 1 2 1 3 1 3
## [38] 3 3 1 2 1 5 4 4 5 2 1 1 7 5 1 0 2 3 1 4 3 3 2 3 3 5 2 4 1 4 2 2 3 2 2 4 3
## [75] 3 1 3 3 2 1 2 2 6 1 4 1 2 4 4 2 0 1 2 3 3 1 5 1 2 2#Promedio de productos defectusos por lote
Promedio_defec<- mean(defectuosos)
#Resultados
cat("Promedio de productos defectuosos por lote:",Promedio_defec)## Promedio de productos defectuosos por lote: 2.57Interpretacion
Simulando los 100 lotes encontramos que, en promedio, hay unos 2 a 3 productos defectuosos por cada lote de 50. Esto coincide con lo esperado, ya que con una probabilidad de 5 % lo normal es que alrededor de 2.5 productos salgan defectuosos por lote.
4. La demanda diaria de energía (en MW) sigue una distribución normal con media de 100 MW y
desviación estándar de 15 MW. Simular la demanda de un año (365 días) y calcular la probabilidad
de que un día supere los 130 MW. y realizar el histograma.
#Semilla
set.seed(2804)
#Parametros
media<- 100
desviacion<- 15
n<- 365
#Simulacion de demanda diaria de energia (en MW)
demanda<- rnorm(n,media,desviacion)
demanda## [1] 105.40086 113.23791 95.66191 77.89918 79.42319 90.81774 108.19144
## [8] 104.68368 100.09306 87.20884 105.15260 104.13989 98.31963 120.35978
## [15] 102.83529 115.27776 96.79373 110.01441 106.11160 106.65044 99.08566
## [22] 124.03312 118.88051 95.20529 89.20410 119.95129 77.67888 110.19558
## [29] 111.49790 110.13048 107.25186 125.24979 114.44995 111.12533 98.39135
## [36] 96.36983 111.51725 104.47528 105.90959 99.06830 93.55826 127.36248
## [43] 114.06115 93.65110 115.85012 75.42752 97.58562 102.19470 119.54433
## [50] 100.49934 69.04352 73.64725 116.41778 109.30212 110.99204 112.04918
## [57] 107.01286 98.15856 91.00234 96.56884 75.98629 107.73186 117.48395
## [64] 64.37215 107.54507 95.75664 89.74181 112.71656 99.32174 119.18918
## [71] 89.70230 114.18205 81.49915 104.59730 100.71918 107.33624 95.13296
## [78] 94.14949 99.90240 87.74861 104.23418 116.51919 98.86748 99.07519
## [85] 102.15609 94.03427 109.05483 86.09211 103.53228 101.82508 86.96238
## [92] 106.41099 68.79638 111.02506 73.70678 83.20245 104.65561 86.67692
## [99] 93.78013 88.92636 118.28787 80.59984 118.52154 105.27032 96.20485
## [106] 85.31337 92.02088 123.00733 107.29998 104.61689 114.39912 120.23916
## [113] 113.04548 113.70845 104.15149 76.54347 109.77034 83.15286 100.10273
## [120] 95.79188 116.79885 102.81586 106.07169 98.57775 99.75475 104.79842
## [127] 105.75555 117.34439 109.52750 107.22980 93.59555 92.45561 105.49521
## [134] 88.37653 82.25955 90.46220 112.20684 88.54297 86.02440 113.61241
## [141] 109.40779 106.52681 109.56455 131.36991 90.94823 125.08761 117.47488
## [148] 109.46030 120.34352 114.05613 105.31198 110.97192 80.47102 136.16810
## [155] 118.51556 124.61412 96.92716 114.21066 98.76957 81.50971 83.84816
## [162] 110.86760 90.42576 92.59424 79.41035 90.04145 116.02068 110.53163
## [169] 99.73436 96.03943 96.22816 121.08774 86.12025 113.80447 98.40710
## [176] 90.14003 85.85866 102.90599 108.15154 115.41783 86.47123 111.55328
## [183] 81.06931 76.90216 105.50451 93.73448 88.52630 87.93353 90.96142
## [190] 63.58264 119.18125 100.78388 114.65778 105.49564 118.30241 97.87179
## [197] 132.18319 88.78794 91.02229 112.27693 115.31935 96.27899 123.08726
## [204] 85.59000 101.87812 104.54933 99.90453 91.40527 108.19015 97.75206
## [211] 75.84574 81.12168 104.85255 106.04516 64.02813 109.05900 74.50026
## [218] 94.61246 106.18603 90.19719 109.81278 96.64549 117.57704 86.51603
## [225] 101.99557 114.73167 105.45526 95.87044 95.96573 119.43926 73.22012
## [232] 93.51455 80.41700 96.05687 115.78116 87.90879 84.08981 75.27138
## [239] 104.08278 120.04548 94.67945 94.09228 91.98189 89.49963 109.37581
## [246] 97.64122 104.01258 126.38264 100.69323 95.94155 107.68451 126.85987
## [253] 89.85442 126.87628 93.45597 99.71036 77.02823 86.77868 98.66183
## [260] 115.99890 107.06618 123.51082 92.23770 117.94488 121.04886 123.30159
## [267] 89.86697 104.65234 128.43098 77.71517 109.84750 98.91375 115.68621
## [274] 131.05556 88.03680 101.29222 61.58668 100.23903 93.88133 112.71647
## [281] 128.82146 90.19319 106.99771 116.73757 104.60596 105.23877 84.82457
## [288] 76.19304 100.54638 95.16018 111.78651 124.54674 97.48557 73.48793
## [295] 108.84305 116.09274 96.34840 80.47953 99.72570 77.86226 86.98309
## [302] 98.14910 96.06597 106.08502 102.52120 133.86862 105.95125 102.02889
## [309] 97.25203 96.79025 95.42594 100.69507 112.77778 105.23074 106.90443
## [316] 97.07880 134.20792 94.58700 110.79270 110.15601 88.42114 66.98304
## [323] 90.03823 108.60498 82.73106 117.26239 107.89061 103.04561 89.13250
## [330] 93.73646 133.78409 99.12645 121.86402 115.67506 96.93785 85.11002
## [337] 104.21511 115.09251 85.90470 97.89581 106.75518 121.63175 101.92372
## [344] 110.36826 95.54140 93.24117 109.63298 97.59370 126.57933 102.00147
## [351] 91.99038 147.46293 104.63130 88.05724 97.49760 75.83904 113.46876
## [358] 98.12225 107.75827 103.64008 112.09677 102.04542 115.70774 93.17482
## [365] 105.79858#Probabilidad que supere los 130 MW
probabilidad <- mean(demanda>130)
#Resultados
cat("Probabilidad de que un dia supere los 130MW:",probabilidad)## Probabilidad de que un dia supere los 130MW: 0.02191781#Histograma
hist(demanda,main="Histograma de la demanda diaria",col="blue",xlab="Demanda (MW)")
Interpretacion
Simulando la demanda de energía durante un año, encontramos que la probabilidad de que en un día se superen los 130 MW es bastante baja (en este caso de un 2 %). La grafica muestra de una forma aproximadamente normal centrada en 100 MW, que coincide con lo planteado.
5. Una empresa de manufactura electrónica quiere simular el tiempo de vida (en horas) de un nuevo
modelo de capacitor. Basado en datos históricos, se ha determinado que el tiempo de vida sigue una
distribución exponencial con parámetro β = 1000 horas, que representa el tiempo medio de vida de
los capacitores.
a) Generar 1000 tiempos de vida del capacitor aplicando el método de la transformada inversa.
b) Estimar la media y la varianza de los tiempos generados y compararlas con los valores teóricos.
c) Graficar el histograma de los tiempos de vida simulados junto con la densidad teórica de la
distribución exponencial.
d) Calcular la probabilidad de que un capacitor dure menos de 940 horas usando la simulación.
#Semilla
set.seed(2804)
#Parametros
beta<-1000
lambda<- 1/beta
n <-1000
# a) Generar 1000 tiempos de vida transformada inversa
uniformes <- runif(n)
tiempo_vida <- -beta*log(1-uniformes)
#Se transforma a la inversa y esto equivale a -beta*log(uniformes)
# b) Estimar la media y varianza y compararlas con los valores teoricos
media <- mean(tiempo_vida)
varianza <- var(tiempo_vida)
# Valores teoricos
media_teorica <- beta
varianza_teorica <- beta^2
cat("Media:",media)## Media: 1006.545cat("Media teorica:",media_teorica)## Media teorica: 1000cat("Varianza",varianza)## Varianza 1079669cat("Varianza teorica", varianza_teorica)## Varianza teorica 1e+06# c) Histograma
hist(tiempo_vida,probability=TRUE,main="Tiempo de vida",col="yellow",xlab="Horas")
#Densidad teorica
curve(dexp(x,lambda),from = 0,to= max(tiempo_vida),add = TRUE)
# d) Probabilidad de durar menos de 940 horas
probabilidad<- mean(tiempo_vida<940)
cat("Probabilidad de durar menos de 940 horas:",probabilidad)## Probabilidad de durar menos de 940 horas: 0.611Interpretacion
Simulando los tiempos de vida de los capacitores encontramos que se acercan bastante al valor esperado de 1000 horas y la varianza también coincide con la teórica. La grafica nos muestra bien la forma de la distribución exponencial, y cuando calculamos la probabilidad de durar menos de 940 horas encontramos que es un poco más del 60 %. En general, los resultados de la simulación confirman lo que predice la teoría.