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.

fallas<-rpois(150,3)
fallas
##   [1] 2 1 2 3 2 2 3 1 2 3 1 3 5 5 4 0 4 0 2 4 3 2 6 3 6 4 2 1 2 2 8 4 1 2 4 5 2
##  [38] 2 2 1 2 1 0 5 2 4 3 4 1 1 4 2 3 4 1 2 4 4 2 5 3 3 2 3 4 3 3 2 5 3 1 3 2 4
##  [75] 2 3 4 3 3 2 4 2 1 6 2 5 4 1 1 4 3 3 6 5 1 5 1 3 5 6 2 6 2 1 2 5 3 3 3 3 0
## [112] 6 0 2 2 4 4 1 3 3 5 4 2 3 1 3 0 2 1 1 1 3 3 2 3 5 2 2 2 4 5 6 1 4 3 1 3 6
## [149] 5 5
hist(fallas,main = " sistema de produccion ", col=2)

media <- mean(fallas)
desviacion <- sd(fallas)

cat("La media de las fallas simuladas es:", media, "\n")
## La media de las fallas simuladas es: 2.893333
cat("La Desviación estándar de fallas simuladas:", desviacion, "\n")
## La Desviación estándar de fallas simuladas: 1.593387

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

vida_util<-rexp(1000,1/500)
vida_util
##    [1]  272.686296   23.260738  232.096596  548.099096   15.156210  939.307286
##    [7] 1665.819716   48.763013 1819.118271 1638.895707  979.056259  747.891277
##   [13]  433.736119   66.219391  316.094844  555.760260  121.358170  287.364700
##   [19]  261.827459 1813.687819  518.270666   59.456758  675.880870  296.538356
##   [25]  355.865812 1394.642211  216.728344   38.578799  185.377513  271.157654
##   [31] 2118.746901  149.423941   59.779382 1111.975693  659.858104  915.028623
##   [37]   72.303586 1476.753032  211.358309   12.567038   25.869677  803.211206
##   [43]  204.758079   52.947708  332.094941   25.409543  511.386068  108.294982
##   [49]  282.404145  122.630823  242.844331    2.874125  172.091867  127.995156
##   [55]   21.898960 1559.077915  516.439470 1570.582197  457.577356  616.896193
##   [61]   25.787067 1943.444009  167.597066  205.043996  780.847934   70.022505
##   [67]  802.274430  265.318879  354.937414  317.424416  669.130802  831.571655
##   [73]  677.428782  674.608904  383.608970    2.046393   67.417571  214.731067
##   [79]    4.833786  121.154272  291.399576   98.087692  278.990622  149.797959
##   [85]  185.106978   75.474047  318.885961  691.939284  281.806965   63.491166
##   [91]  244.840774  255.059040  196.316559  297.195252  298.246385   24.928552
##   [97]  129.610866   54.968623  334.180192  212.970844  105.848079  323.279609
##  [103]  598.509704 2793.720655  104.612839  284.402563 1447.014131  447.116445
##  [109]   55.932026  200.103053   62.308853   35.521591  114.235550   12.010926
##  [115] 1420.105990  214.033060  540.048579  143.490668  168.629005  189.777990
##  [121]  157.222569   63.955992  651.748890  573.581640  214.442173  622.447451
##  [127]  863.256894  453.725695  129.480302  127.247550  122.378646  109.223111
##  [133]  547.146416   85.126760   17.932842  536.130662   38.584697  244.781240
##  [139]  201.757167 2454.680019  367.864968  370.170886  399.875805    8.486948
##  [145]  771.590041  299.110404  556.404348  164.405147 1046.281064  890.944175
##  [151]  733.049930  528.432312  395.592044  425.158800  356.193545  883.119319
##  [157]  175.694674  420.269360  724.225914  163.102364   49.048828  474.514338
##  [163]  926.266885   47.474422 1014.475665  631.672430    4.694010  377.256046
##  [169]  112.115841  307.979442  403.654613  218.715027  614.767030  175.653377
##  [175]  303.476248  327.589700  611.256116  297.165012  532.888723  201.102993
##  [181] 1477.926526   55.706753  509.121572  278.674853  435.614084  656.843342
##  [187]  516.219140 2020.823422   88.501247    2.698342   31.212628  980.811634
##  [193]   12.430045  933.592388  606.482670   77.135209  119.583488  105.606382
##  [199] 1872.738171  200.945082 3157.563320  139.714638  240.297252  445.675453
##  [205]  449.102148  108.747388  301.901878   48.083712  241.544222  722.516599
##  [211]   25.376303   92.386318   59.070692  144.915166  393.746030   44.542442
##  [217]  212.355650  974.568106  139.288564 1859.966504  371.233100  331.039277
##  [223] 2109.572107  446.668186  632.443381 1486.671276  249.785673  577.510372
##  [229]  661.617374 1782.734212   10.482105 1446.500000  247.961669  526.090782
##  [235] 1440.674229  169.176369    8.883958 1909.705301  117.280162   22.804003
##  [241] 1325.881177  351.328031  185.747996  963.226483  622.968219  636.695511
##  [247]  666.055446  362.582635  633.436223   92.084536  419.829404  517.283489
##  [253]   86.113479   17.254276  107.758975  331.467705   38.070470  196.081876
##  [259]   84.813594  802.671265  456.812050  175.561378  111.175502  995.267084
##  [265]   16.582613  699.801799 1168.600875   81.147549   49.878022  590.587640
##  [271]  545.356331  227.057737  390.478099   73.917124 1276.434177  426.314607
##  [277]  268.954215  719.042102  293.552381  742.113556  454.472655    6.298159
##  [283]   17.391428   75.270276  262.074858  476.619913  248.578339 2815.824620
##  [289]  330.544773  445.510425  741.895715 1164.579386  360.928459  283.102650
##  [295]  415.672279  113.377553  308.333712 1112.789912  417.061168  393.592229
##  [301]   47.436110   61.436545   60.418533  993.635343 2029.141868  292.543884
##  [307]  634.256788  444.464735  241.875282   90.844264  341.658197  610.614847
##  [313]  830.740673  491.595406  521.900959  234.590683   96.791198   23.027090
##  [319] 1174.156182  884.837180  177.215905  412.132479  175.547501 1886.358396
##  [325] 1354.924730  449.732894 1449.072402 2352.625767  187.851183   89.247943
##  [331]  362.774489  973.837196  495.285522  132.124845  149.521506  281.262773
##  [337] 1540.420309  602.944548  464.347437  962.249709  591.039067  516.442899
##  [343]  237.622127   70.599839  424.609171   81.637556  783.481756  909.432491
##  [349]   23.282917  779.208033   78.438788 2375.801260  352.033142 1562.959049
##  [355]  342.558303  100.776572  335.261028  762.552886  147.617691  454.957052
##  [361] 2766.547405  394.050214  331.476892 1988.542060  273.723481  323.620623
##  [367]  337.903473  816.871420  123.938082  251.129237  346.779227  181.323155
##  [373] 1153.674576 1256.715200  784.825608  209.165122 1665.004242  316.426824
##  [379]  432.058889 1165.093428 1123.124764  235.838421  292.045126 1281.871475
##  [385]  231.402154  463.929302  535.411650   52.656047  270.941782 1024.326161
##  [391] 1293.984550  488.465283 1067.942017  289.933195  489.647194   41.432386
##  [397] 1252.196799  472.564483  245.127191  152.507249  387.993345  427.436155
##  [403]   75.662380  708.751435  154.860052  386.534295  528.226153  753.788457
##  [409]  110.280963  221.208172  281.546594   35.999855  133.265499  284.410276
##  [415]  686.434836  637.238198   89.839537  562.429258  498.944793   99.608238
##  [421]   87.262970  218.948250  206.434336  469.469638 1830.937798  757.453668
##  [427]  661.184307  373.063194   78.558003  772.606191 1514.243377   43.728139
##  [433]  275.476828  294.653296  592.922274  743.798036  832.415483  428.771079
##  [439] 1270.497088  267.503871  764.914032 1187.636261  391.171337  474.526006
##  [445]  535.304692 1871.649506  161.810329  490.911764 1044.688658  896.860192
##  [451]  977.076977  619.652386  942.413090   90.569493  703.223844  409.104649
##  [457]    2.603043  237.095848  212.301244  630.315032  913.174885   93.526213
##  [463]   50.257286 1108.038354  546.862349  141.966624  285.933347  142.662098
##  [469]  727.715132  306.185820   84.111698  533.421422   32.694210  238.946992
##  [475]    7.822502 1614.594155  315.440304 1412.823368  909.480395  465.815539
##  [481]  381.659433   67.511978  199.361279   93.450270   75.364392  303.779857
##  [487]  501.468425  318.068004   11.301653  427.857600  384.339791   66.199738
##  [493]   80.767251   10.449016 1716.170549  825.223435  562.250623   13.567224
##  [499]  754.522464    3.046404  540.651517  676.428453  283.389780  421.660230
##  [505]   81.455101  755.505783  888.558054   76.253188  252.139119  382.043859
##  [511]  117.256846  228.533622  769.956405  837.092111  272.208183  106.609312
##  [517]  180.116905 1305.930267 1176.552695   99.515672 1304.923301 1050.154671
##  [523]   26.960716  222.202316  202.835917  171.746027  801.822934  123.938642
##  [529]   49.578962  271.605841   61.990621  465.295769  420.605612  300.817800
##  [535]  208.581439 2183.508260   56.413837 1418.437174 1213.410572  423.283772
##  [541]    1.743821  378.193513  210.457788  140.582000   17.660719 1429.541521
##  [547]  352.902480  706.630900   64.438638  717.467249   15.743523  422.745538
##  [553]  248.348933 1060.292947  120.616653  448.006507 1328.543561  976.859075
##  [559] 1667.535953 1168.911616  942.178846  746.470704  486.575565  660.664657
##  [565]  282.502274  356.942703  729.279390   58.896161  317.753076  954.679970
##  [571]  112.498127  342.971866 1783.587861   51.371634   13.051480   50.724312
##  [577] 1123.958842  640.718789   10.730166 1582.813927   53.274981   47.134748
##  [583]  270.196869  666.248587  446.034462  329.016635  654.482999  237.727627
##  [589]  353.419142   51.133158 1428.040914    3.932333   42.611307  250.828301
##  [595]  683.724448  862.071244  375.093996  223.139343 1014.490852  209.269993
##  [601]  940.927272 1022.832959  320.927807   22.738633  504.009310   70.906077
##  [607]   41.661295  545.212942  751.638089   48.479084   23.392207   51.723111
##  [613]  975.384598  176.167410   52.503573  181.676006  174.811479  162.435046
##  [619]   96.292898  261.663727  118.484827  993.469050  941.464804  804.362681
##  [625]  149.484521  421.535828  156.009187 1005.166858   45.911919  429.578261
##  [631]    4.732408   41.872174  930.228167  380.039477  348.757584  686.393849
##  [637] 1712.041331    7.499231   45.278979  496.061808  809.427640  364.276329
##  [643]  248.043941  513.790532  836.697707  316.605040  689.271431  119.713925
##  [649]   33.597431  383.530813  200.763786 1111.053601  126.734135   41.040799
##  [655]  108.914842  608.614321  376.639816  429.921423 1319.571773  385.363082
##  [661]  381.197826  180.626792  286.307721  353.066961 1532.471299    5.057269
##  [667]  273.675745 1008.204543  260.393516  389.732475  311.411454  416.158910
##  [673]   60.168401 1096.513105   20.033766  165.698735   52.973805  169.902962
##  [679] 1374.951555  615.747507  339.633372  188.762890  129.111187  723.578064
##  [685]    9.461518  896.463879  447.768684  402.024763  673.043413    3.057887
##  [691]  106.400263  150.220726  445.928475  568.997629  128.677122  246.538006
##  [697]  120.339660  579.876848 1028.531009  175.511085  559.681622  907.801055
##  [703]  544.833132  132.724678  452.263730 1127.345565  192.527274   13.239547
##  [709]  321.696930  257.147155  933.460949  266.210217  140.497288 1270.622673
##  [715]  812.907924  732.469040   83.450598  946.237267 1186.889690  275.459401
##  [721]   87.209811   89.065416  712.666349  339.852975 1022.928677  187.415986
##  [727]  533.786034  224.112794  579.181720 1469.463918  138.849074   17.066269
##  [733]  530.566413  285.041156  241.272815  367.393217  758.783514  791.512155
##  [739]  614.446566  494.778105  100.842476  594.912545  696.105638  433.596388
##  [745]  165.140261 1708.121978  348.634850  659.161908  873.596798  183.450202
##  [751]  632.820554  182.718580  379.001480 1376.234491  203.014088   37.859601
##  [757]  350.409264   10.096350 1029.948497  251.706730 2115.347469  273.541556
##  [763]  328.505261  601.278498  256.285598  251.742350  261.735331 1066.909714
##  [769]   18.380967  318.920856  571.864671  701.791718  236.281766 2300.433496
##  [775] 1000.010456   47.687050   88.661064  244.762774   53.584471 1393.733858
##  [781]   74.298262  515.557506  964.379996   62.961222   18.109108  414.167189
##  [787]   14.993666  540.309939  442.571372 1181.868350   20.350098  207.687245
##  [793]  462.320356  902.322338  217.640188 1861.890103  136.835330   89.410260
##  [799]  627.077708  464.874571  106.403428  166.787899   24.818068  420.275649
##  [805]  251.515285   78.483350  651.636161  385.538477  277.023168 1071.524342
##  [811]  106.857454  343.938252  205.220350  333.970164  111.038561  149.371933
##  [817]  113.731072  185.845851    9.997576  493.051903  368.279968  763.870616
##  [823]  269.949983 1194.156982  229.681293  131.239284  160.524898  869.725590
##  [829]   53.978471 1194.539044    1.000710  386.781442 1175.184156  861.347679
##  [835]  831.718468  913.515138  502.472636  968.949875  721.867350  215.284113
##  [841]  450.709821  842.848779 1258.792560  787.460742  165.563234  121.577208
##  [847]  262.264239 1041.845194  306.634108  220.005707  210.383332  744.554801
##  [853]  434.858951   16.884076  805.171643  672.898187  121.174713  584.880243
##  [859]  374.315441 1388.858142  995.987330 1349.201606 1399.794415 1110.410885
##  [865]  404.912251  787.773434   23.733493   58.656133  854.634773  200.813673
##  [871]   46.040215  718.460703   68.510227  451.233234  190.237842  268.062164
##  [877]  307.860390  791.702203  130.077792  221.697297  646.366948  133.150804
##  [883]  734.338425  509.392572  224.182905  470.789642  407.364332  117.920186
##  [889]  161.892084  162.309460   47.112050  147.965560  440.466438 1038.234216
##  [895] 1153.626791  381.652581  466.000579  180.664040   91.439897  815.013699
##  [901]   81.974705  838.210825  110.399093 2168.723462 1266.451939  219.074149
##  [907]  458.627439 1000.333556   19.527333  518.582271  810.061069  234.088510
##  [913]  115.828395  697.870401  158.324025  262.574252   87.458835   82.424869
##  [919]   12.945518   31.770132  337.565991 2058.649386  133.852999   52.694064
##  [925]  443.068055  426.446978  369.177992 1070.617446  320.976139 1511.838119
##  [931]  212.172763  259.340575   67.335560  340.674808  355.841674  239.071262
##  [937]  315.153644  116.856330   66.890903  880.956477   50.789304  372.694558
##  [943]   55.450126  103.216053   14.809564  361.522738  457.663091  247.251958
##  [949]  944.999926   60.549360 1054.217593  123.753233 1726.538990  722.712896
##  [955]  289.159650  431.419376  599.104632  643.279512  500.629766  125.100231
##  [961]  302.628474  750.373942  158.274757  870.713507   62.664338   32.620471
##  [967]  584.223909  377.756679   47.377281  852.755608  112.649664  847.864994
##  [973]   51.441307   50.354552  117.788196   15.962306  223.532485   36.402847
##  [979]  648.316985  645.931215   74.820196   76.346732  311.885377  404.614351
##  [985]   30.689138  556.974542  120.455504  619.542417  806.455619  428.094378
##  [991]  196.786841  205.833095   99.310212  550.716559 2370.490396    3.662711
##  [997]    4.581976  131.739055  376.581972  861.954045
hist(vida_util,main = " vida util de un componente electronico ", col=4)

probabilidad_simulada <- mean(vida_util > 700)
probabilidad_teorica <- exp(-1/500 * 700)

cat("La probabilidad simulada de que dure más de 700 horas:", probabilidad_simulada, "\n")
## La probabilidad simulada de que dure más de 700 horas: 0.245
cat("Probabilidad teórica de que dure más de 700 horas:", probabilidad_teorica, "\n")
## Probabilidad teórica de que dure más de 700 horas: 0.246597

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.

ensamblaje<-rbinom(100,50,0.05)
ensamblaje
##   [1] 2 5 1 1 1 3 0 4 3 2 1 1 2 0 4 4 1 3 2 3 3 2 1 3 4 2 6 1 1 3 3 1 3 2 3 1 5
##  [38] 4 4 2 2 3 5 1 2 1 2 2 3 1 3 2 3 3 3 1 1 1 3 4 4 2 2 3 3 1 3 3 4 4 4 5 1 1
##  [75] 5 5 2 2 1 4 2 3 3 2 3 4 2 3 0 3 1 2 1 1 4 2 2 4 1 1
hist(ensamblaje,main = " linea de ensamblaje ", col=6)

promedio_defectuosos <- mean(ensamblaje)

cat("Promedio de productos defectuosos por lote:", promedio_defectuosos, "\n")
## Promedio de productos defectuosos por lote: 2.46

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.

demanda<-rnorm(365,100,15)
demanda
##   [1]  93.61624 111.36570 107.53015 132.05644  88.06494  88.98363  92.68946
##   [8] 108.66714 102.64977  97.42541  90.69724 102.38339 100.38273 111.01138
##  [15] 121.59303 110.68057  95.13306 110.39989 110.85545 117.04329  96.39302
##  [22] 116.17617  62.88868 107.30162  94.70933  96.07440  69.83813  77.43206
##  [29]  99.95814 108.94883 126.30282  76.75294 103.66525 120.36011  83.61661
##  [36] 127.77398 100.90634 108.05167  99.19363 107.88570 102.04712 103.11405
##  [43] 111.71274 130.13925  98.41729 111.46145 103.35113  95.62093 108.91099
##  [50] 103.44757  74.67935 107.53058  89.76431  97.29431 110.03798  94.44508
##  [57]  91.04614 113.03632  87.43824 123.49352 130.05951 104.56102  77.41989
##  [64]  75.34561 107.23104  93.27193  82.53915 103.46074 101.61277 122.33312
##  [71] 129.63497 110.64307 125.91576  72.17886 105.01111  98.81266 103.51243
##  [78]  88.43460  86.97366  99.47202  93.82245  94.11291 146.17910  90.24364
##  [85] 104.35902  73.94676  68.26391  88.79323  92.38773  92.20382  85.32467
##  [92]  97.57242  83.55699 118.58427 102.65832  78.40519 104.13771  71.64375
##  [99]  87.31996 104.45238 113.18653  82.94600  81.02280  73.66715  94.42271
## [106] 104.78009  96.61032  99.21064 119.65596 101.06991  81.68830 108.46650
## [113] 108.26465  97.15631  93.87876 122.03751 102.40418  87.12129  78.45659
## [120] 122.01528 118.60236  94.34096  99.78650 118.19910  87.95270  87.35620
## [127]  88.96162 106.71887  80.94146 110.15532 100.77498 115.15204  85.08910
## [134] 105.13243 136.18315  99.64703 124.22585 112.98243 117.45142  73.06713
## [141] 110.18376 102.85697  89.18906  88.28024  99.70406 120.59618 116.86416
## [148] 122.32233 118.67860  98.19503 120.49958 102.63036 111.72125 123.44522
## [155]  98.68036 125.85352  98.84002  95.78437 106.13307 111.08848  92.69201
## [162] 108.50788  83.56753  76.58672 111.30411  89.45741  83.91149 100.19919
## [169]  95.05453  71.24847  93.97240 109.99561 103.62457 107.19963  81.40779
## [176] 131.85189  89.44958  99.26985 122.92500  87.50630  91.08777  82.28092
## [183]  99.64946 109.50132  78.23122 103.65583  92.63900  92.52565  82.87584
## [190] 112.16043 110.02189 119.11941  64.52474 103.50936  99.83210  80.11578
## [197]  75.56985  98.79990 119.82076  83.15642 105.28022 106.29427 113.27649
## [204] 107.89027 139.18763  94.97789 104.52452  69.73011  65.62459  75.29962
## [211]  91.50823  85.20917  93.84968 101.00745 105.97150 124.18762 104.84416
## [218]  89.62837  64.20129 116.04863  90.31747 115.04051  99.74025 122.10717
## [225]  93.87635 100.86610  97.97609 117.04622  92.97632 106.99964  92.84501
## [232] 129.17491  83.57269  98.23362  98.85054  87.66526  92.67047 111.55412
## [239] 108.33960  92.00486 110.28578  84.84969  78.74134  97.94637 125.03667
## [246] 107.82218  92.78947  87.45957 102.92079  74.61940 111.78728  77.28196
## [253] 123.09175 112.50881  82.55195  83.16544 102.87142  91.84837 111.10391
## [260] 106.08053  96.53579 116.37452  71.28565  96.82717  86.65703 100.56087
## [267] 114.05048 108.48926 112.53955 100.75571 131.01124  89.00464  96.73321
## [274] 117.42818 123.06306 106.19703 107.97867 100.45109 116.84578  79.77479
## [281] 101.89627 118.26812  87.90339  96.69799  88.38175  76.08675 115.02270
## [288]  99.11642  88.63758 112.96820  89.70488  98.97896 113.71654  80.97319
## [295] 121.53788  80.25591 116.65648  97.39031  95.05540 127.41882  90.56004
## [302] 102.80906 102.46294  60.29687  98.15478  71.86396 102.91907  95.14336
## [309]  88.94560 116.52082 103.79892  74.34778  91.56183  94.36359  83.81207
## [316] 117.28737  95.99451  91.30501 103.07032  96.64430 103.57847 112.27211
## [323] 104.15664  81.42742  65.10855  92.08079  87.58094 104.17397  94.93424
## [330]  98.24500  78.69552 107.97677  79.23771 116.34868  91.65228  87.64249
## [337] 106.02313 100.61885 120.98450  73.80330 107.24148  81.51905 117.12743
## [344] 121.33504 109.03849  93.54546 110.56302 114.97438 124.54474  55.85947
## [351]  87.61516 103.49092  86.54074 134.88717 101.93125 113.41610 113.15962
## [358]  94.85270  99.91498  97.59200  84.86798  95.22548 122.92801 107.98637
## [365] 107.35033
hist(demanda,main = " demanda diaria de energia ", col=8)

probabilidad_simulada <- mean(demanda > 130)

probabilidad_teorica <- 1 - pnorm(130,100,15)

cat("La probabilidad simulada de que la demanda > 130 MW:", probabilidad_simulada, "\n")
## La probabilidad simulada de que la demanda > 130 MW: 0.02465753
cat("La probabilidad teórica de que la demanda > 130 MW:", probabilidad_teorica, "\n")
## La probabilidad teórica de que la demanda > 130 MW: 0.02275013