Ejercicios

(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.

n <- 150
tasa <- 3
poisson <- rpois(n, tasa)
poisson
##   [1] 1 3 5 4 3 4 4 2 4 0 1 4 4 3 2 2 3 4 2 4 5 4 2 4 4 3 7 3 3 2 2 1 5 2 3 1 0
##  [38] 5 1 2 4 0 1 1 1 3 3 1 2 4 3 6 5 4 6 2 6 3 5 4 3 2 2 2 4 5 3 4 2 3 2 1 3 1
##  [75] 3 2 1 1 6 1 4 2 6 3 3 2 5 1 3 3 5 2 2 1 0 6 5 7 5 4 5 0 3 5 2 1 0 5 2 8 3
## [112] 2 1 3 1 4 2 2 3 0 2 2 0 2 3 4 4 0 0 4 1 1 1 3 1 8 2 1 4 3 3 2 7 3 2 3 2 3
## [149] 2 4
media <- mean(poisson)
desv<- sd(poisson)
media
## [1] 2.873333
desv
## [1] 1.738999

(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 <- 1000
media <- 500
exponencial <- rexp(n, 1/media)
round(exponencial,1)
##    [1]  429.7  619.8 1461.6   75.3  467.5  583.4   45.4   53.9  484.7  322.1
##   [11]  129.6  522.7 1143.4  125.4    9.7  149.3  385.6  248.8 1643.5  186.3
##   [21]    6.6  646.3   30.8   47.8  178.1  134.2  461.7  386.3  140.2   91.1
##   [31]  472.7  152.0  459.3   71.4  181.7  394.2 1847.6  271.2 1046.6  163.5
##   [41]  739.0 1097.5  143.9   68.5  628.4  907.8  363.0  542.3  663.1   18.1
##   [51]   70.9 1532.9  861.3  274.1  814.6  979.9  479.3  127.6  315.2  191.4
##   [61]  461.3  515.2   18.3  181.0  438.9  169.1  507.5  467.7  396.1 1793.9
##   [71]  246.3   31.6  379.2   12.0  257.7  252.1  332.3   88.2  496.3  539.5
##   [81]  384.2  408.7  507.3   27.0  976.1  565.3  239.7  418.4 1018.4  591.5
##   [91] 1157.4   60.0  227.2  113.2   96.6   28.7  809.9  106.5 2181.4   92.1
##  [101]  209.7 1076.4  437.7  167.2 1201.9    7.5  415.3   25.2  524.2  290.7
##  [111]  754.3  169.2  326.9  104.7   32.7   71.3   31.1 1279.8  169.6  438.3
##  [121]  660.9  616.8  204.7 1040.0 1057.3  215.8  411.6  130.7  155.4   99.0
##  [131]  220.8  290.4  963.8  763.3  242.8  307.5  851.9   68.3  245.2  101.3
##  [141]  241.3   82.7   89.1  753.6  191.1  314.0  229.2  360.7  483.8  411.0
##  [151]  432.9  367.1   68.7  319.9  524.0  679.3   67.8 1038.6  706.2   79.4
##  [161] 1034.6  330.5  235.0  509.2  177.6   42.5 1297.5  623.3  135.5  759.8
##  [171]  674.8  719.3  865.0  450.6  537.8  458.5  203.2  377.2   86.6  546.2
##  [181]  122.3  402.9  224.3  652.1  267.1  921.1  146.1  212.4  844.8  672.4
##  [191]  875.9  187.1    7.8  398.3   62.0  173.9 1012.2  240.0  100.6  940.3
##  [201]  842.3  651.1  329.8  713.5   13.8  387.3 1223.8  197.9 1887.0  455.3
##  [211]  295.8   28.6  302.5  479.9  592.6  406.1  619.7   60.8  901.4  404.3
##  [221]  992.8  134.4 1065.1  217.2  118.2  803.9  236.0  201.4 1023.4  118.5
##  [231]  469.0  170.6  420.7  621.1  356.2 1426.4 1694.5 1001.2  708.9  445.9
##  [241]  755.2  570.7  437.3  360.3   26.6  230.3   15.4  511.0   69.6 1927.0
##  [251]  564.4  576.9  997.0   85.8  253.9  579.4  572.5  621.5  394.9  832.6
##  [261]  419.0 1811.7  272.6  940.5  420.1  292.4 1190.7 1491.4 1082.0  847.1
##  [271]  443.1  117.7   66.7  317.2  183.2  261.8  201.8  139.3  285.2  303.9
##  [281]  265.7  737.5  586.5  104.0  258.5  418.4  238.0  515.7  318.4  947.7
##  [291]  328.3  110.4  140.7  170.8  154.7  174.7  317.5  123.8  122.4  207.9
##  [301] 1668.8   15.8  235.8   19.1 1796.7 1348.3   57.6 1117.1  711.3   54.2
##  [311]  119.7  516.9  458.4  221.5  176.0  251.0  176.0 1614.1    1.3  466.2
##  [321]  524.9  503.6  546.1  220.3  171.1  170.4  743.2  454.2 1339.3  154.1
##  [331]  712.8  599.8  639.8 1261.4  185.2  256.5  286.9  421.2  999.8   28.8
##  [341]  508.6  333.3  324.1   55.3  893.5  574.0  618.6 1070.8  105.3   58.3
##  [351]  493.2  813.0  437.0  364.4  795.3 1076.0 1225.6  240.1  187.3  533.7
##  [361]  209.7    4.8 1823.7   18.7 1655.6   88.2  276.4 1646.5  791.3   85.8
##  [371]  758.9  143.8  168.6  119.4 1069.2 1035.7  989.6 1069.2 1034.4  220.2
##  [381]  199.9  358.4  433.7 1495.7  131.3  138.5  128.2  256.5  209.2  891.3
##  [391]  292.1  557.2  494.5  173.5  423.5 1004.4  734.6  106.6  386.8  101.2
##  [401]   18.4 1056.2   89.0  219.8 1162.6   24.1  370.4   85.2  653.5  121.1
##  [411]  307.3  111.5  701.3 1613.1   22.1 1693.1  355.5    4.2  225.6 1170.0
##  [421]  178.5  161.9  788.6   76.3 1046.0  586.2  116.0  198.3  146.1   72.1
##  [431]   59.2 2880.2 2634.6   53.6   40.3  343.5  179.0  835.5  483.7   40.5
##  [441]  612.1   40.0  134.4  864.4 1480.0  834.3  101.0  446.7 1084.5  230.7
##  [451] 1062.1  170.1  103.6  120.6  243.8  284.6   19.4   23.1  154.0  164.8
##  [461] 1136.7  579.6 1246.7  124.1  663.0  190.8  223.8  448.9   83.3  255.2
##  [471]  494.7  700.3  277.9   29.7  126.7  504.3   21.4  393.0   62.8 1826.0
##  [481]  138.1    7.9  139.4  409.3 2302.8   53.7    4.1  145.0 1045.7  488.4
##  [491]  559.7  128.8  711.6   66.4  604.4   66.8  603.9  560.3  360.0  347.9
##  [501] 1180.7  567.0 1956.3  122.4   93.7  341.5  168.2  439.6  357.3  210.6
##  [511]  609.4 1846.0  336.5  451.3  150.2  914.2  171.3  455.3 2659.3   56.4
##  [521]   42.8  266.2  303.2  211.1  376.3  254.0  197.9  756.7  261.2  633.9
##  [531]  334.0  158.8  140.6  201.9  141.1  123.3  657.5  177.0  346.7  107.8
##  [541]  272.7 1135.8  304.1  110.0    9.0   24.0  241.3  511.5 1346.9   23.2
##  [551] 1048.3  639.4  227.5  167.6  153.7  630.2  275.6  168.3  959.5 2980.7
##  [561]  135.7  579.9  697.5 1462.9  103.4 1900.7  137.4  356.7  160.6  658.6
##  [571] 1775.5  893.7  399.7  446.9 1620.4   22.1  133.7   33.2  355.2   25.5
##  [581] 1739.8  157.5 1761.3  185.8  506.5  405.5  700.1   56.1  653.3  453.9
##  [591]  148.7 1975.6  290.2  579.9  449.8  190.8   12.4   48.6 1373.0   24.7
##  [601]  350.9  963.9    0.9   63.5 1306.1  633.7  274.1  812.0 1931.9  129.8
##  [611]   53.8   44.4  792.6   88.3  213.8   74.4  596.1  176.1  143.7  174.6
##  [621]  180.4  826.5  265.3  201.6   38.8  475.4   34.8   34.6  704.4  431.8
##  [631] 1215.3 1121.0  211.5  145.0  184.7  176.5  104.2 1322.8  246.3  290.0
##  [641] 1173.6  118.7   94.4 2688.8 1023.2   69.3   22.4 1510.4  218.0   61.0
##  [651]   18.7  178.7  819.1  292.5  979.2  742.8    0.0 1568.5 1487.0  113.6
##  [661]  197.0 1770.2  362.9  108.8  862.2  404.6  941.7  478.4  352.1  151.8
##  [671]    6.5  643.0  302.8  267.0 1366.5 3840.4  505.9  152.1  219.1  651.9
##  [681]  732.0 1148.2  191.8  349.1   58.2  483.5  282.4 2007.4  958.0  127.6
##  [691]  199.1  933.6  407.2  287.3  200.6   37.3  432.9  568.3  552.4  254.5
##  [701]  146.6  105.6 1365.1  178.4  751.0  182.3  252.3  281.8  368.7  352.4
##  [711] 2038.7  586.3   14.9  244.9 1600.1  807.8  495.6  547.1  680.9 1400.1
##  [721]  844.7  403.1   90.8   74.9   65.9   63.6  115.7  146.6  679.9 2016.6
##  [731]  485.8  549.0  558.3  680.7  202.8  541.1  301.8   29.0  239.0  110.7
##  [741]  142.8 1030.7  606.1  145.8  927.9  274.4 1741.0 1246.2  135.9  342.5
##  [751] 1275.7  192.6  474.6 1803.1  324.6 1350.6  426.2  891.1  182.8  181.4
##  [761]  540.9  562.0   48.2  203.9  149.3  665.3  438.4  454.3 1340.0    1.9
##  [771]  446.5  261.0   61.7  144.9  400.1  392.6 1330.5 1166.3  103.2  328.0
##  [781] 1401.3  134.9 1999.5   79.9  602.1   89.0  380.8    2.0  627.5   58.9
##  [791]  878.3 1492.3  566.2  171.9   72.1  497.1  125.2  168.7  339.0  963.4
##  [801]  827.1  211.1  723.1  490.7  405.7  212.7  110.3  152.0   46.7  447.5
##  [811]   92.9  629.3  690.4  194.5  890.8    6.0   11.7    9.9  173.4   64.9
##  [821]   20.4   73.8  650.5  210.5  199.3 1020.7  235.6  359.6  568.9  351.4
##  [831]  235.4  176.1 1362.3  479.0  806.7  335.8  583.0  880.2 1016.4  174.3
##  [841]  378.0  428.9  805.2 1339.3    9.7 1489.2  867.3  415.8  109.9  401.7
##  [851]  116.5   64.5 1155.3  115.7  552.7  173.1  456.6  257.2  262.4   48.7
##  [861]  334.9  889.3 1354.8  170.2  662.3  403.3   70.1  794.7   36.5  480.4
##  [871]  406.5   32.5  745.2  112.9   54.0  361.9  823.7  103.7  223.6  645.5
##  [881]   30.7  289.5  304.2 1010.7  508.0  351.4  890.0  160.7  245.7  846.1
##  [891]  988.0    0.9  274.2  884.7 1330.1  232.8   23.3  184.7  389.9  254.1
##  [901] 2497.2  344.9 1587.8  460.9  473.8  388.9 2231.7  611.3  123.7  251.7
##  [911]  158.8  805.9  663.6  460.0  147.3  293.5   14.6  328.3  674.6   40.7
##  [921]  395.4  939.4 2188.9  358.0  174.0  425.9 1145.3  318.6  537.1    9.9
##  [931] 1603.3   28.1   76.3  192.7  684.6 1014.5   32.4 1028.3  682.3  290.3
##  [941]  254.8 1088.7  478.6  239.0  279.7  196.6  413.3  244.9  797.4  238.4
##  [951] 1156.9   23.9  389.7 2773.2  693.2  538.5    3.0   74.9  373.6  225.0
##  [961]  124.8  167.1  547.1  990.0  361.6 1442.4   56.5  438.5  143.0  823.0
##  [971]  321.3  680.6  422.7 1507.7  278.2  310.4   99.9   88.2  183.9   37.1
##  [981] 1031.1  177.8   73.5  672.9  589.2   91.3  837.2  522.7  181.0   61.5
##  [991]  244.2  197.0   44.8  311.2   52.8  381.7  206.3  517.0   54.5   52.4
prob <- sum(exponencial > 700) / n
prob
## [1] 0.237

(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.

n <- 100
lote <- 50
probabilidad <- 0.05
binomial <- rbinom(n, lote, probabilidad)
binomial
##   [1] 2 2 4 3 3 1 1 1 1 1 4 2 5 2 2 4 2 2 4 1 4 4 0 3 4 2 8 0 0 4 4 3 4 5 5 1 3
##  [38] 4 1 2 0 0 2 2 1 1 4 4 3 2 1 2 4 0 2 3 4 1 2 4 1 3 5 4 2 3 2 2 2 3 0 2 3 0
##  [75] 2 2 2 1 4 3 2 3 2 5 1 1 2 1 2 1 1 5 0 4 5 2 3 1 3 4

(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.

n <- 365
media <- 100
desviacion <- 15
normal <- rnorm(n, media, desviacion)
prob <- sum(normal > 130) / n
prob
## [1] 0.02739726
hist(normal)

(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. Simular 1000 tiempos de vida del capacitor aplicando el metodo de la transformada inversa.
n <- 1000      
beta <- 1000
exponencial <- rexp(n, 1/beta)
round(exponencial,1)
##    [1] 2618.9 3031.6  152.7  944.2   76.7  309.8  992.8 1133.9  254.1 2211.6
##   [11]  109.6  214.4 1714.3  807.9  117.9   58.9  645.7 1257.5   39.0  162.9
##   [21]  819.8 1566.5  103.5  334.6  862.3 1387.5  260.0 1655.9  586.0  545.1
##   [31]  796.3  538.7  236.1   15.1  379.2 1522.2 2737.7 1774.2  308.5  902.0
##   [41]  829.8  939.9  147.1 1765.9  570.4  169.3 1671.1 1927.6  355.8  189.0
##   [51]  703.8  138.4  488.8  967.7  173.9 2215.2  508.3 1507.3  844.8  330.8
##   [61]  510.8 1391.8 1536.6 3040.0 1976.5 1819.9 2566.7   19.3   55.9  478.3
##   [71] 2391.0 1158.6   52.8  300.0  528.0  207.6 2244.5 1500.7  792.1  754.3
##   [81]    7.4  601.5 1711.1  523.8 1961.6  792.0  605.1  450.9   64.0  825.5
##   [91]  364.0  271.4   16.1    5.3  212.4  309.1  394.5  117.6  320.2  700.5
##  [101]  321.9 2117.4  188.1 1103.5  393.0 1115.5   37.5   53.9 1911.2  161.3
##  [111]  226.1  147.2 2655.1  176.8  726.5   49.4 2667.6 2782.5 1580.0  288.5
##  [121]   36.7 1372.4  355.0    1.7 1048.4  413.1  239.8 2495.5  185.9  330.4
##  [131]  380.5 2867.4 1304.0  184.7   49.6  123.3 1975.9  151.3 1381.1 2672.7
##  [141]  443.4  259.5  145.3 4357.4   36.8 2021.6  340.3  731.5   88.7  207.8
##  [151] 1377.6  676.7 1900.1  812.6 3136.0  998.3 1178.6  757.8  557.8  352.4
##  [161]  416.6 1011.5  771.9  152.4  982.1  674.5  828.6  780.4 1396.2   17.6
##  [171]  469.7  650.0  335.4 2284.3  726.5  132.2  651.4 3133.7 2388.6 1011.8
##  [181] 1136.3 2454.9  575.8 1933.9 4006.9  371.7  936.7   77.0  120.4 1039.5
##  [191] 2131.5 1720.4   54.3 1960.5  344.3   62.8   69.8  978.9 1068.0 1510.3
##  [201] 5075.2  858.3 1512.9 1813.3  563.5   70.4 1171.2 1638.9  129.1  298.9
##  [211] 1541.2  840.1 2845.6 3246.7  952.6  356.7 1084.4 1922.2  137.4  311.9
##  [221] 1303.0  741.2  386.5 3210.0 1197.0 2228.9 1888.3 1178.2 3176.3  210.7
##  [231]  929.6 1921.5 1921.1  742.1   79.2 4620.0 1145.1  198.2 1632.7  166.2
##  [241]  116.9  941.5  349.8  635.1   10.5   64.9  250.3  319.0  363.1  289.2
##  [251]  785.9 1437.4  257.3  684.2  605.2  583.6  704.4  270.3 1750.0 1447.0
##  [261]   83.9 1860.0 1526.2 1504.0 2667.6 1506.9 1084.2  623.0  494.6    1.1
##  [271] 2217.7 1487.3  227.4  687.2  664.8 1114.8  281.4  266.1 1778.1  246.8
##  [281] 1075.9 1228.0   12.4  293.5 1407.0  513.2  755.6  445.8   21.2 3364.0
##  [291]   64.4 2026.0 1417.6  536.4 2236.6  572.3  415.5  554.7  173.5 1983.1
##  [301]  466.5  599.2  199.7  437.0  100.1  527.8 3061.8 4417.7  267.7 1790.3
##  [311]  239.2  218.3  794.8  401.2   72.1 1252.8  263.0 2111.6   32.7  722.7
##  [321] 1062.7 1971.7 1584.7  205.6 1812.2 1930.7  181.5  734.8 2559.1 1684.2
##  [331] 1868.3 2193.8   19.8   90.7 1513.9  470.3 1336.1 1395.1 1103.3 1136.0
##  [341]  962.5  268.8 1201.2  538.5  756.8   29.3 1848.1 1177.6  363.3 1429.3
##  [351] 1495.3  625.9   63.6  308.9   28.7   56.9 2607.7  489.5  241.3 2478.0
##  [361]   16.9 1067.4  381.9  907.5  457.8 1900.0  633.4  337.8 2053.5   56.8
##  [371] 2422.6  999.6   58.2  671.4 1316.9 2659.4  582.1 2229.0  464.9  686.1
##  [381]  275.0  438.1  201.0 1901.0  969.9 1429.6  713.5   11.3  452.3 1076.1
##  [391] 1273.9  380.0 2894.6  849.4 1834.1  289.2 2887.0 3092.0 1312.7 1258.2
##  [401]  198.6  982.6  872.8 1108.9  840.7 1362.5  330.8 1064.6 1032.8  995.5
##  [411]  221.6 1385.2 1153.2  340.1  423.7  951.0  261.5 1006.0  167.8  760.7
##  [421]  511.0 3397.3  207.0 2094.3 1733.8  659.9   29.3   86.6  333.4  166.2
##  [431]  963.5  933.9  245.3  366.3 1166.9 1598.9  422.4  578.7  432.0 1809.1
##  [441]  653.3  656.9 2242.4 1203.5  339.2  538.0  371.7 2663.8  185.7  475.3
##  [451]   92.3   17.6 1652.8  418.2 1504.0  254.0 3093.5 1287.0 1309.6 1199.5
##  [461] 2595.2  430.9 1414.7 2248.4  663.2    2.4   29.0  680.5  212.5 1178.8
##  [471]   73.2  456.9  304.4 4230.7 1858.1  230.8   99.6  666.3  830.3   86.6
##  [481]  107.5  740.8   89.9 1891.6  508.5  367.1  922.1  461.2  384.2  293.0
##  [491]  180.6 1512.8 1015.5  426.0  525.7  143.9  619.1 1525.7  997.7  419.5
##  [501] 1874.9  125.3  654.8  887.0 4100.7  122.2 2561.1  871.2   36.1  145.6
##  [511] 1215.3  268.5  306.7  367.2  366.4 1018.2  500.8 1708.6  200.8  622.4
##  [521]   67.0 1362.3   23.6  778.3  357.3  527.0  791.1  900.3 1168.4  490.3
##  [531]    8.1 1583.6 1648.8 1542.5  355.4  300.8 1964.7 2360.3 1095.4 5387.3
##  [541] 1378.4   17.6  954.8 2638.6 1875.7 1854.7  959.8  654.3  162.6  889.3
##  [551]    8.9  577.4 1607.4  213.1  430.9 2336.3  378.6 1174.2 2742.5  335.1
##  [561]  112.9  354.6 1058.4  362.1 1652.7  285.4  213.0   82.4  805.8  479.9
##  [571]   42.0  244.9  804.8 1619.7  787.5  191.4 1083.5   36.5 1037.9 3061.6
##  [581] 1478.8  827.4 1012.7 3510.5 1592.3 3047.3  424.4  685.6   34.8  427.5
##  [591]  612.2  195.4  168.2 1938.9  319.8 2178.7 3211.5  389.4 1132.6  933.7
##  [601] 1532.9 1180.3 2328.1 2235.2 1209.5  519.4 2194.9 2233.6  470.8   83.2
##  [611] 2539.5  177.8  257.9  357.7 2700.2 1481.4  339.1 2252.7  350.5  800.3
##  [621]   46.4  889.4 1047.2  389.3  889.7 2268.7  358.4 1290.9 1451.0 1116.6
##  [631]   94.1 3000.6 1294.2 4252.2 3802.8  779.0 3211.8  225.6  992.0  812.1
##  [641]  762.0 2520.7  807.8  173.9  822.6 1409.5 1053.2 1064.3  365.0 1154.2
##  [651] 2312.2  652.4 1558.9  675.3   72.6  552.3  553.3  369.8  569.1  743.3
##  [661] 1686.3 1356.7  309.3   16.4 2085.3  554.6 1819.8 2842.1  181.8  875.1
##  [671]  578.0  584.3  416.6  437.5  970.1 1296.2  302.6 3757.5  401.6 1845.0
##  [681]  129.9  612.6  410.1  867.0  419.6 1349.9 2687.7  338.6 2434.3  978.1
##  [691] 1717.2  191.0  744.5 1496.7 4523.4  563.9 1975.8  534.5  672.2  319.9
##  [701] 1547.2 1383.8 1798.2  267.3   26.2  861.9  323.8 3352.3  700.2 3038.4
##  [711]   41.2  171.7  979.6 3092.7 1025.5  227.9 2501.1  517.7  183.4  292.0
##  [721] 1437.2  125.6  704.2 1074.6  741.9  616.6 1546.9 1693.3  166.1   39.6
##  [731] 2746.9  155.2  380.7   99.3  698.4  813.2 2187.7  372.5 2237.3  305.3
##  [741] 3526.1 1660.0  762.7  471.9  321.7  125.5  708.4   51.3  783.3 1078.4
##  [751] 1239.3   96.9   54.7  938.4 2088.2  748.1  199.4  169.0  795.2  447.8
##  [761] 2143.8 2880.0  193.2  385.4  886.2 2611.1  493.3 1641.7  696.3 1030.3
##  [771]  320.9  531.5  914.8   39.1  210.7 1688.4   23.0  648.9  415.5  517.8
##  [781] 1445.2  127.6  485.9  573.0 2126.1 1114.2 1700.8  762.2  200.0  457.4
##  [791] 4088.5 2196.4  140.7  596.2  573.3  780.0  787.3  247.0 1943.4 1724.6
##  [801] 1294.4  387.4 2359.0  171.1 1737.6 1255.8 1746.3 1019.6 2314.1 1781.0
##  [811] 2578.7   45.0  905.3  466.9 2096.7  824.7  827.1  782.7  310.2 2420.8
##  [821]   35.7  609.1  971.8   22.5 2826.8  277.6 2703.0   81.5  277.5 2013.5
##  [831]  308.7  445.7 1094.0  851.4 1091.7   83.4  412.9  272.0  101.4  396.3
##  [841]  153.7  270.8 4577.4 3047.6  216.5  986.5  240.7   91.6 2299.5  853.3
##  [851]  434.3  176.0  186.5  757.7 3703.3  874.2   16.7  807.5 2207.4    5.0
##  [861] 2557.0  172.8 1707.2  678.2 1409.6  245.5  279.7  412.1  688.1 1127.8
##  [871] 2623.3  605.2  445.8  550.1  567.8  386.2  800.2 3625.2 1753.7 1896.3
##  [881]   10.1  354.7 1551.5  247.0   59.7 1103.3 2509.5  953.2 3522.8   90.9
##  [891]  208.5  468.6  928.4  105.7  314.8 3162.0 1505.1  642.6  327.0 2039.2
##  [901] 1457.4 3139.5  246.0 2642.4  723.7  662.4 1540.2  147.9  220.9 2215.5
##  [911]  400.0 1539.4  239.9 1644.9  663.2 1346.3  668.0  575.9  510.0 2685.0
##  [921]  998.4  206.1 3088.5  719.2 1395.5  919.4 3389.3 1548.0  620.2  201.4
##  [931]  102.3 3351.0 1840.4 2784.3 1067.0  700.6  558.4  841.3  625.0  191.3
##  [941]  204.7   80.3 1092.4   53.5 1023.2  723.0  479.4   81.9 1112.5 3073.5
##  [951]  689.8  750.1 1795.5 1282.6   82.4   25.6 1985.6 2126.9 5035.1  297.8
##  [961]  408.0  443.6  685.4 2153.8  829.9 2098.7 1682.3  238.8 1306.0  949.5
##  [971]  904.9  225.8  718.0  477.6   75.0  615.1  924.8  891.6 1198.5  218.2
##  [981]  204.2  377.7  805.7 1192.2 5850.1   77.2 1380.0 1961.8  627.5  284.0
##  [991]  536.5  527.1  373.9  661.2  343.8  346.7  244.8 1272.2  550.6  259.2
  1. Estimar la media y la varianza de los tiempos generados y compararlas con los valores teóricos.
media <- mean(exponencial)
varianza <- var(exponencial)
media
## [1] 1001.176
varianza
## [1] 879701
  1. Graficar el histograma de los tiempos de vida simulados junto con la densidad teorica de la distribución exponencial.
hist(exponencial, breaks = 30, probability = TRUE, 
     col = "skyblue", border = "black",
     main = "Tiempos de Vida Simulados",
     xlab = "Tiempo de Vida (horas)")

curve(dexp(x, rate = 1/beta), col = "red", lwd = 2, add = TRUE)

legend("topright", legend = c("Densidad Teórica"), col = "red", lwd = 2)

  1. Calcular la probabilidad de que un capacitor dure menos de 940 horas usando la simulación.
Prob_simulada <-sum(exponencial <940)/n
Prob_simulada
## [1] 0.601