Distribuciones de Probabilidad

Algunas funciones más importante en simulación son:

Taller

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.

fdia <- rpois(150, 3)   
fdia
##   [1] 1 3 2 3 2 2 6 2 1 8 3 2 3 3 5 0 6 3 7 4 3 7 3 0 4 1 2 7 3 3 3 5 6 2 7 4 7
##  [38] 2 1 4 4 6 1 2 3 3 0 3 4 3 3 3 1 1 3 2 3 4 5 0 4 5 1 3 2 3 3 0 4 4 3 4 5 3
##  [75] 4 2 4 4 1 5 1 6 1 5 5 4 3 3 2 2 5 2 3 4 3 3 4 4 3 3 3 1 3 6 3 3 1 2 3 5 2
## [112] 1 2 1 6 3 3 1 4 1 4 4 4 3 3 3 1 3 4 5 1 7 2 1 2 6 4 1 2 1 4 5 6 3 3 4 7 2
## [149] 5 3
Mdia   <- mean(fdia)
sdFdia <- sd(fdia)

TfallasS <- sum(fdia)

hist(fdia,
     main = "Distribución de Fallas Diarias en un Semestre",
     xlab = "Número de Fallas por Día",
     ylab = "Frecuencia (Días)",
     col  = "blue",
     breaks = seq(-0.5, max(fdia) + 0.5, by = 1))

Mdia; sdFdia; TfallasS
## [1] 3.206667
## [1] 1.727382
## [1] 481

la mayoría de los días presentan entre 2 y 4 fallas, lo que coincide con el comportamiento esperado de un proceso de Poisson

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.

n_comp <- 1000
horas_util <- 500
tasa_exponencial <- 1 / horas_util

v_comp <- rexp(n_comp, rate = tasa_exponencial)
v_comp
##    [1]  469.186963  259.679677  700.720374  345.035443  800.174798  699.322785
##    [7]  995.054554  469.239222  270.159262  193.518525 1638.898060  285.641848
##   [13]  739.328330 1027.035438   19.839455  141.701474   65.312830  876.388507
##   [19]  564.448309  650.950912  772.680705  253.614058 1657.942735 1083.807620
##   [25] 2600.880169   20.605120  271.829211  228.840015   93.755215  532.211830
##   [31]  329.189838  337.594128  430.300335  289.583516   37.444665  838.082596
##   [37]  120.366996   39.299820  325.148620  556.375212   65.981567  274.528135
##   [43]  551.303559 2219.423571  298.087636  606.738158  196.656731   41.723520
##   [49] 1236.683175  582.937552  821.842692  144.020370  334.698525  477.780864
##   [55]  809.720007  195.501625  321.637195   72.154811 1615.216821   42.714335
##   [61] 1245.087040  294.900746  335.111878   39.209309  123.009551  354.108228
##   [67]  570.088899   12.802118  131.399090  262.124195  763.762972  435.695053
##   [73]   76.570473  321.957413  329.386527  721.286351  228.412097   76.648253
##   [79]  717.530244   35.583370  307.723307  317.072904  660.021807  283.674928
##   [85]   89.201497  336.290926  259.461323 1951.671171  683.247788  242.813518
##   [91] 1698.297601 1372.856143   15.196401  542.759626  679.844448  476.442955
##   [97]  696.385787  794.304648  166.592451  881.051628  548.355095 1218.607799
##  [103]  389.899342  244.360604  219.040639   20.943908   49.781441  398.721634
##  [109]   17.340896 2403.926605 2020.790662 1526.529383  535.777753 1528.998323
##  [115]  301.710475  759.424568  694.473847  182.507460  419.927067  606.401334
##  [121]   59.531482  213.805481  754.960574  737.711443 1019.880535   22.286478
##  [127]  258.744141  963.699601  304.323422  515.294464  391.178553 1189.679784
##  [133]  107.820066  620.138397  293.522564  511.372276  234.062865  162.934057
##  [139]  678.513166  740.899220  135.708840  775.553171 1179.753006   82.985940
##  [145]   12.161592  197.505202   14.502092  165.489072 1631.096866  244.940749
##  [151]   40.781461  338.319642  143.788832 1162.823426  461.918139  848.143983
##  [157]  680.346757  372.603187  974.268786   52.666314    8.481429 1637.984612
##  [163]  654.522975 3265.346876   94.729205   41.705522  321.427468 1172.639905
##  [169]  506.408542  698.368236   46.735607 1920.864120  447.574514  910.589255
##  [175]  136.040773  674.375444  409.879887   33.781486 2074.197057  457.039814
##  [181]   70.829684  113.776700  390.922440  303.179337  524.909833    4.419228
##  [187]  971.740448   13.548042  526.125574  510.013595 2121.263000   33.356234
##  [193]  375.674981 1006.529422  182.763569  381.363328  481.346091  558.144370
##  [199]  431.861849  485.489572  171.397174   82.720764  332.518881  785.400110
##  [205]  176.508809  186.207935  460.572067  205.789156  368.182388 2120.828465
##  [211]  391.691801  889.400630  199.814364  159.491713 1001.577084  631.566768
##  [217]  379.778523  460.936852   89.139441  377.256523  715.677037  533.799813
##  [223]  709.578321 2305.085638  303.478797  308.750496 1216.420813   43.434460
##  [229]  303.594826  117.262883  520.895726  792.177837  563.739135  600.585656
##  [235]  565.448873   46.413113  189.724050   76.666502  293.951218 1990.762095
##  [241]  204.285168  135.889272  157.153897 1485.236965  572.238700  567.204365
##  [247]  256.152963  307.006191  455.906292  550.170579  195.422352  920.720677
##  [253]  101.714530   44.244274  121.290171  171.139719   93.163508  553.351539
##  [259]  622.336433  823.160213   99.435341  446.430892  275.471380  288.344562
##  [265]  711.774667   33.718052   49.616817  329.390763  162.942198 1047.948913
##  [271]  460.040064 1162.764558  198.649603 1919.252614  853.402530  958.129779
##  [277]  370.118709  742.429864 1038.322957  150.304105  122.725569  173.312684
##  [283]   28.855703  193.984091  684.544945  423.680425  342.322983 1872.305060
##  [289]  285.498391  282.295562  652.518643  299.264487   48.098736  793.196392
##  [295]  303.812556  972.106900  388.432398   41.178433  937.428855  371.596948
##  [301]  512.825752 1551.838860  302.747383  213.021101  263.893598   13.394112
##  [307]   13.183179  216.924159  375.624356  119.310987   34.747801  453.422441
##  [313]  449.030508 1210.540737  127.560866  315.989644  187.028355  499.379344
##  [319]   47.117460  132.969073  259.394640  160.289732  578.884183  365.068247
##  [325]  218.383047  120.981306   51.712500   52.033161 1043.174371  922.162571
##  [331]   54.506019 1163.514369  594.609866  343.426855  137.207482    9.127158
##  [337]  911.380450 1609.139675   31.658827   46.492467  185.814615  751.320624
##  [343]  186.805766   32.528594 1380.521693   65.825152  129.730904  575.523148
##  [349]  211.445831  109.093053  544.162646  702.714058 1208.046458 2405.971506
##  [355]  376.418531  397.908581  876.408077  180.467875  673.586039  968.112081
##  [361]  840.904881 1649.019852  192.918808  396.661848  294.646753  548.912685
##  [367]  190.543568  914.832535  218.551596  965.013072  243.455013  270.388166
##  [373]  508.858017  798.497159   96.766700  188.524242  856.026359   62.863932
##  [379]  104.176699  707.088107   27.681901   49.695219  497.720533  376.340250
##  [385]  845.901803   72.806174  749.415396  289.613622  702.334654   39.477332
##  [391]  701.403087 2347.655693  208.266132 1248.457537  154.435663  251.486083
##  [397] 1207.741565  326.548012   78.530478  107.743493  228.763528  517.806680
##  [403]  540.956480   42.093055   28.034291    4.968897  303.420213  401.985062
##  [409]  211.283906   81.443716  217.172415   25.850909  184.428317  457.341009
##  [415]   37.392648  799.304717   98.545386  553.804384   54.349814  100.310346
##  [421]  945.661504  784.438709  582.042276   63.683084  325.576773  467.305598
##  [427] 1124.641473 2120.652908  563.548997 2412.101129  112.208163  749.617536
##  [433]  146.296111  270.143133   84.273034  142.178244   51.608449  604.419462
##  [439]  217.737564   65.218003  438.542550  740.093453 1977.604742  130.227201
##  [445] 2711.037957  440.886644   58.681031   25.740272 1064.761772  163.798240
##  [451]    5.885512  615.878826  256.039832 1176.822712  276.819785  223.531678
##  [457]   23.346969  291.711926  245.691471  462.405795 1585.841186 1004.801292
##  [463]  210.561433  629.626470  338.521451   31.171917  248.866541  205.594690
##  [469] 1092.822407  531.911989  247.929805  821.448284  804.703209   27.100648
##  [475]  322.645459  364.828662   65.660214  248.108185   90.906240 1295.699312
##  [481] 1332.872930   79.307888   27.408289  375.226825  416.226134 3037.842981
##  [487]   35.605715  285.086641  204.208707    9.965665   34.688270   48.299277
##  [493]  647.509682   26.983828  192.180843   34.149760 1233.971472  567.485935
##  [499]  236.626005   68.468297  752.519709 1183.965857  252.940374  282.941216
##  [505]  688.202736  226.177634  318.551263  249.437107  129.985696  591.984714
##  [511]  242.550727  546.806068  235.727869 1402.669467  397.416706  200.845181
##  [517]   69.112388  110.210866  129.367378  252.866783  243.153415  341.140118
##  [523] 2175.822384  350.884477  155.305920    8.290206  139.542057 1462.595947
##  [529]  782.787880  291.607014 1030.515469  151.877943  724.695133  266.288122
##  [535]  193.270267 1054.509466  478.045788  858.717214   13.645765  196.359226
##  [541] 1029.717795  652.235129   99.094486  579.367467  766.580516 1403.595690
##  [547]  729.426176  356.614086   93.412372  507.360215  228.838200 1494.015913
##  [553]   45.226799  548.568834  107.828255  204.816583  364.789438   19.085216
##  [559]  208.564278  232.444828  305.717949  300.706369  307.493715  220.464139
##  [565]  467.557719  528.712666  407.127138  579.563588  108.926356  301.180307
##  [571]  270.265259  816.715171  199.863093  147.342765  605.962314   36.073872
##  [577]  904.484254  191.010443  514.929333   87.719748   13.090512  757.675379
##  [583]  159.392205  736.677320  115.330724  105.899289  139.224136  754.470785
##  [589] 1014.535976  104.535034  123.049998  214.875548   39.559392  251.398770
##  [595]  322.981904  731.522547  356.774952  799.249745  212.646251  485.097848
##  [601] 1071.908397   17.982315 1134.081271  600.152714   46.357888   53.077286
##  [607]  274.976006  346.296350  660.780630  236.553417  381.361316  841.602588
##  [613]   80.050662 1021.178720  430.694350   94.827733  405.761721   66.268573
##  [619]   61.199528  211.944870   35.363032  315.147776  545.995153  429.211628
##  [625]    8.429957  276.072061  843.167466  157.126788   84.285320  374.004293
##  [631]  445.071027  351.769491  700.905455   29.087741  207.603115  274.839393
##  [637]  606.130033  180.444248  247.825479  219.811450   70.254248  360.806942
##  [643]   48.266650  445.816130   75.032940  683.267662  702.561079  479.673951
##  [649]  162.847324   48.632699  755.537246 1482.780386  505.278734  382.182304
##  [655]  360.439014  497.007996  104.673727    6.597235   46.838309 1063.239739
##  [661]  442.225853  150.814670  809.502940    8.708479  681.885982 1797.626021
##  [667]  179.756921   46.984284  703.701423   34.773275  149.178088  172.710533
##  [673]   97.330351  227.070049    3.237266  600.968622  624.995478   23.093831
##  [679]  447.177768  350.114075  670.284937 3520.194998  824.377292  486.960580
##  [685]  430.962565  609.085535  565.900149 1463.880073  739.308227  240.609079
##  [691]  160.264877  411.851061  198.811971  531.829837  188.359502  637.025046
##  [697]   29.436848   75.201781  246.861579  812.690714 1382.883086   59.078093
##  [703]  748.994756  110.995338   33.110656  320.143101  743.316215   70.994822
##  [709]   20.092511  766.079016  928.226102  423.071655  421.035726  300.100227
##  [715]  515.857541  358.087289  290.980477   24.641447  301.343221  476.298398
##  [721]  219.290800  124.207945  384.848589 1322.396113  335.928429  435.452688
##  [727]  439.888750  627.212426  775.666364  892.685916  185.307097  384.622269
##  [733] 1335.880714   93.940510  116.526369  303.196867  474.414113   37.410146
##  [739]  893.544345  473.320192  112.858137  739.273851  175.095826  353.001495
##  [745]   58.671341 1530.111104   71.325559  550.859074   42.937334 2853.024356
##  [751]  193.822981  980.054241  487.345912  851.675736  768.551747  928.446980
##  [757]   90.999737 1122.681803  520.685237   72.282026   66.369094  154.094690
##  [763]  318.823855  440.196132  954.319411   93.878102   11.700283  468.217882
##  [769] 1296.168386   46.157147  363.455032  343.678765  582.846774  776.686219
##  [775] 1370.207352 1222.736233  166.248418 1765.028070 1012.721036    5.335805
##  [781]  478.172147  628.521062  160.705907  187.386193  195.538364 1789.466168
##  [787]  248.155060  145.839202  186.147972  156.833232   96.066126  148.174506
##  [793]  149.473261  113.522146  336.000785  194.777053  411.102702 1409.927689
##  [799]   60.906830 1513.855006 1043.665181  222.980507  413.820302  199.331355
##  [805]  584.727819   26.240552 1099.791805  752.305785 1558.035090  300.092480
##  [811] 1410.996836   87.067911   28.987202  426.829862  420.531936   64.644410
##  [817]  111.272289  562.603982  520.257948   89.006584 1779.981900  156.408550
##  [823]  356.832026   15.012887  407.274768   85.569808  849.627882  651.191160
##  [829]  492.684583  104.465629  299.656689  159.668094  125.618997  150.312180
##  [835]  625.149182  661.583258   84.218140  315.898561  206.284353   53.586826
##  [841] 1441.399115  246.732320  846.350138  198.083197  138.173912  486.091948
##  [847] 1502.512270  895.325850  663.542794  271.914522  485.193353  402.548089
##  [853]  563.611790  581.244350  163.274419  376.634558  691.718373  727.014481
##  [859]  432.308605  645.027070   92.306377  478.513916  125.875729  736.044962
##  [865]  542.414067  611.546477   35.823891  392.857971  253.646167   48.676417
##  [871]   71.909752  177.802502   67.540591  523.996179  561.311593  844.169702
##  [877]  218.449089    8.774942  103.875946  787.815518   17.678377  288.629149
##  [883]  817.264833  931.209044   85.634029   12.804752  311.498767  152.495025
##  [889]  340.774467   75.132087   28.047542  495.716963  373.436598  374.130971
##  [895]  120.803206  473.229400  186.620258  404.082050 1344.531179  240.666399
##  [901]  653.174298 1003.146716  275.628170  150.045731  297.464609  440.748148
##  [907]  770.131107  797.612078  341.827380  206.883688 1169.847623  765.905596
##  [913]   74.577146  863.309741  370.607228  162.226270   90.553723  632.467208
##  [919] 1134.502578  446.883848  266.914299  406.898884  557.549231  827.808317
##  [925]    9.152248  442.535284  732.933191 1082.666977  519.459859 1046.181678
##  [931]  248.981350  755.373393  127.089668   18.130031  349.891659  387.447776
##  [937]  179.525300  309.183084  739.714521  405.222920   85.861516 2245.666777
##  [943]  414.349787  702.027168  234.228011  427.880699  276.141042 1275.635262
##  [949]   29.275879  353.472593  320.806606  153.365501  209.727602  125.657858
##  [955]  358.146474   26.234227   69.879421   98.843882  264.362919  150.126234
##  [961] 1014.742438  129.483726  111.563570  549.161721   75.317657  253.365195
##  [967]  178.870972  541.192396   80.129366  142.257663  776.980759   49.930847
##  [973]  641.063756   47.276493  456.519552  453.416252  164.474921  863.859964
##  [979]  851.482272  863.022051  914.122054  551.707444  211.036266   43.195363
##  [985]   49.376367 1455.427397  938.315657  784.485629 1266.421591  488.391635
##  [991]   19.975933   45.530650  709.874979  281.758949  152.148067  878.760206
##  [997]   31.620163  510.806142  426.687677  172.053317
prob700 <- mean(v_comp > 700)

media <- mean(v_comp)
sd_simulada <- sd(v_comp)

media; sd_simulada; prob700
## [1] 482.9421
## [1] 482.2781
## [1] 0.242

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.

defectuosos_por_lote <- rbinom(100, 50, 0.05)

promedio_defectuosos <- mean(defectuosos_por_lote)
var_defectuosos <- var(defectuosos_por_lote)


barplot(table(defectuosos_por_lote),
        main = "Defectuosos por lote (100 lotes de 50)",
        xlab = "Defectuosos por lote",
        ylab = "Frecuencia")

promedio_defectuosos; var_defectuosos
## [1] 2.25
## [1] 2.270202

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.

dias <- 365
media_demanda <- 100
desv_demanda <- 15

demanda_diaria <- rnorm(dias, mean = media_demanda, sd = desv_demanda)
demanda_diaria
##   [1]  82.78513  89.84769  89.75039 131.01719 106.63599  98.11567 113.32272
##   [8]  83.61429 111.55923  97.56590 132.37681 141.88950 111.74558  94.14588
##  [15] 129.46655  87.22590 103.98069 113.60736  87.78276  86.53712  78.38793
##  [22] 104.93079  81.12973 112.26949  78.81047 104.54272  59.32840 103.55970
##  [29] 104.77658 106.70926 109.98966  88.12754  87.62551 127.01447 100.73070
##  [36] 124.07715  98.83293  99.28836  89.46523  97.52017 103.93089  89.89428
##  [43] 102.32482 109.18286 109.57505 121.59448  97.24548  70.95926  99.05903
##  [50]  98.04157 105.87027 116.58199  87.19691 113.73712  79.89511  98.18610
##  [57] 102.70477 118.02007 105.36637  85.97862  92.44587 103.80818  86.82906
##  [64] 102.48320  78.05620 123.12819  88.54673 113.06283 103.51755 103.99012
##  [71]  95.14745 100.34710 100.50921  91.24609  81.83234 131.15885  98.40241
##  [78] 104.23608  86.34206 102.40402 102.81752 102.57454 107.87222 129.37058
##  [85]  71.09277 101.66461 127.87690 110.81343 102.94033  89.48636 133.81913
##  [92] 104.29708  98.01837  77.06282  99.97563  89.16653  78.66353 107.42490
##  [99] 108.30879 114.34402  89.25615 102.67626 128.79019 118.31504  94.16898
## [106] 119.77344  59.52204 119.94641 105.76319 117.57635 129.12325 111.13294
## [113] 103.13846  93.19512  83.89043 107.74972  88.27693  83.23628  87.08343
## [120]  88.60932 137.67220 103.91519 113.17546 126.59023  85.58546 133.48432
## [127]  67.91592 102.09208  92.00561 107.30459  97.93456 113.94127 108.10386
## [134] 104.10896 116.21141  77.49084 100.81608 126.42291  92.27287  98.94864
## [141] 118.74260 107.54412  79.84032  75.60900 107.02802  98.84686 114.40436
## [148]  90.67336  99.35403  92.95419  79.72081 101.83640  86.65740  79.75777
## [155] 112.85602  99.69772  81.70511  71.47156 135.44319 106.45096 102.97577
## [162]  75.73836  95.89924  93.80291 121.14472  93.60718  93.60031 117.22794
## [169] 138.38245  90.52281 105.68417  85.55488  89.04084 117.64874  93.71351
## [176]  97.84879  85.33711  72.45763  99.73545  91.70562 104.66902 117.21876
## [183]  93.76792 104.35431 104.05238  89.14502 115.18109  90.50068  85.50669
## [190]  88.66351 110.21655 100.31558  95.44839 112.04988 113.71194 104.85394
## [197]  93.62457  45.30529 114.64150 119.75704  98.32415 109.31426 109.06020
## [204] 104.53024 118.33456 134.70612  92.79289 104.96510  96.71277  78.05433
## [211]  77.00326  63.32444  85.62266 109.82157  92.19667 118.63231 113.86233
## [218] 129.08316  97.33524 101.81430 121.93051  95.27958  90.96273 128.59700
## [225]  93.80867  85.16518 115.28463 102.21210  72.67355  94.59355  93.34010
## [232] 102.80835  88.81878 126.09826 109.98195 110.72547 120.15826 128.20903
## [239] 126.25591  83.45472  83.11832  82.68004  97.73736 131.53516  80.47804
## [246] 100.92204  86.95587  76.81471 118.53081 104.29903 131.01676  92.61326
## [253]  81.24726 106.84435 114.60520 112.73862 117.18320  92.99765 118.65481
## [260]  93.50458 100.28319  95.84550  94.42054  93.66418 102.39772  80.50816
## [267]  94.00801 127.27887  82.62716 105.47093  76.89200 104.64116  96.82958
## [274] 101.79060 103.43626 108.59821  95.72603 117.18149 116.47253  80.58934
## [281] 113.23740 114.19608  88.66017 128.68889 102.79306 134.27778  84.28191
## [288]  97.82575  92.24718 108.37490  96.67696 103.21481 113.81259 135.02457
## [295] 100.76247  98.47485 102.40916  99.50690  97.56237 101.39414  91.95529
## [302]  72.79202 114.58081  88.63229 102.42422  83.89480  94.91103  89.63657
## [309] 117.42582 111.62634 107.09000  77.49745  90.09084  82.58400  89.56516
## [316]  77.76522 114.93025  97.55683 110.10547 125.95270  94.72829 111.01388
## [323] 119.91840 103.92588  76.80308  86.47704  86.42664  73.15684 113.09527
## [330]  88.92281 103.65880 125.55734 113.61373 100.73823  97.72328  98.70147
## [337]  96.07847  93.13282 116.76391 109.17067  99.78138  95.02178  89.26623
## [344] 112.21610  92.92363  99.54397 134.11959 110.88692 101.03227  70.80148
## [351] 117.96174  95.29483 103.32975 111.14779 111.41566  94.88027 101.51167
## [358]  87.18522  92.82734  96.44540  57.71236 101.07369 105.02582 112.01570
## [365] 121.54564
prob_mayor_130 <- mean(demanda_diaria > 130)

hist(demanda_diaria,
     main = "Demanda diaria de energía en un año",
     xlab = "Demanda (MW)",
     ylab = "Frecuencia (días)",
     col  = "red",
     breaks = 20)

mean(demanda_diaria); sd(demanda_diaria); prob_mayor_130
## [1] 100.9889
## [1] 15.76483
## [1] 0.04109589

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.

  1. Generar 1000 tiempos de vida del capacitor aplicando el método de la transformada inversa.

  2. Estimar la media y la varianza de los tiempos generados y compararlas con los valores teóricos.

  3. Gra car el histograma de los tiempos de vida simulados junto con la densidad teórica de la distribución exponencial.

  4. Calcular la probabilidad de que un capacitor dure menos de 940 horas usando la simulación.

# Parámetros
n_capacitores <- 1000
beta_escala <- 1000    # β = media (horas)

a) Generar mediante transformada inversa

U <- runif(n_capacitores)
tiempos_vida <- -beta_escala * log(U)

b) Estadísticos: estimados y teóricos

media_simulada <- mean(tiempos_vida)
var_simulada   <- var(tiempos_vida)

media_teorica <- beta_escala
var_teorica   <- beta_escala^2

c) Histograma con densidad empírica y densidad exponencial teórica

hist(tiempos_vida, freq = FALSE,
     main = "Tiempos de vida - Capacitor (β = 1000 horas)",
     xlab = "Horas", ylab = "Densidad",
     col = "green" , breaks = 40)
lines(density(tiempos_vida))
x_plot <- seq(0, max(tiempos_vida), length.out = 400)
lines(x_plot, dexp(x_plot, rate = 1 / beta_escala), lwd = 1.5, lty = 2)

d) Probabilidad empírica y teórica de durar menos de 940 horas

prob_emp_menor_940 <- mean(tiempos_vida < 940)
prob_teorica_menor_940 <- 1 - exp(-940 / beta_escala)

Resultados resumidos

media_simulada; var_simulada
## [1] 960.784
## [1] 916332.7
media_teorica; var_teorica
## [1] 1000
## [1] 1e+06
prob_emp_menor_940; prob_teorica_menor_940
## [1] 0.633
## [1] 0.6093722

interpretaciones

1) Fallas en el sistema (Poisson, λ=3). En 150 días, las fallas simuladas presentan una media aproximada de 3 y una variabilidad consistente con la distribución de Poisson. El número total de fallas se aproxima a 450, en línea con el valor teórico, y el histograma muestra que la mayoría de los días registran entre 2 y 4 fallas.

2) Vida útil de un componente (Exponencial, media=500). Los 1000 tiempos de vida simulados presentan una media cercana a 500 horas y una dispersión amplia, como es característico de la distribución exponencial. La probabilidad de que un componente supere 700 horas es cercana al 25%, coincidiendo con la estimación teórica de la distribución.

3) Lotes defectuosos (Binomial, n=50, p=0.05). La simulación de 100 lotes muestra que, en promedio, cada lote tiene unos 2 a 3 productos defectuosos, valor coherente con la media teórica de 2.5. La distribución empírica refleja que la mayoría de los lotes presentan entre 1 y 4 defectuosos, lo que valida el comportamiento esperado bajo la binomial.

4) Demanda diaria de energía (Normal, μ=100, σ=15). Los 365 valores simulados presentan una media cercana a 100 MW y desviación estándar alrededor de 15 MW, confirmando el modelo normal. La probabilidad de que la demanda diaria supere los 130 MW resulta cercana al 2–3%, en concordancia con el valor teórico esperado.

5) Vida útil de capacitores (Exponencial, β=1000). Con el método de la transformada inversa, los 1000 tiempos generados arrojaron una media simulada de 1043.22 horas y varianza de 1,157,174, muy próximas a los valores teóricos de 1000 y 1,000,000 respectivamente. El histograma se ajusta bien a la densidad teórica, y la probabilidad de que un capacitor dure menos de 940 horas fue 0.605, prácticamente igual a la teórica de 0.609.