II Parcial

librerias:

library(tidyverse)

library(ggplot2)

Puntos

Ejercicio 1 (1.5 punto)

Ejercicio 2 (1.0 punto)

Ejercicio 3 (2.5 punto)

Ejercicio 1 (1.5 punto) - Inciso 1.

1. Crea una función que reciba como entrada n y σ, donde genere un vector de n componentes de muestras aleatorias de Yσ.

muestras <- function(n, sigma){
  
  muestras_de_x <- rnorm(n, mean = 0, sd = sigma)
  muestras_de_y <-  vector()
  
  for(i in 1:n) {
    if (muestras_de_x[i] > 1){
      muestras_de_y[i] <- muestras_de_x[i]
    }else if (muestras_de_x[i] <1){
      muestras_de_y[i] <- 0
    }
  }
  print(muestras_de_y)
}

Ejercicio 1 (1.5 punto) - Inciso 2.

2.Para n = 10000 y tomando σ^2 = 2, calcule el valor esperado de Yσ y la varianza de Yσ. Dibuje el histograma de Yσ, usando ggplot.

Y = muestras(n = 10000, sigma = sqrt(2))

##     [1] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2.983305 0.000000
##     [9] 1.060686 0.000000 0.000000 0.000000 1.084562 0.000000 2.123528 0.000000
##    [17] 0.000000 0.000000 0.000000 0.000000 1.219199 1.641288 0.000000 0.000000
##    [25] 0.000000 0.000000 0.000000 0.000000 1.164339 0.000000 1.701203 1.187183
##    [33] 0.000000 0.000000 0.000000 3.264468 1.223354 0.000000 0.000000 0.000000
##    [41] 0.000000 0.000000 1.001733 0.000000 0.000000 0.000000 0.000000 0.000000
##    [49] 0.000000 0.000000 0.000000 0.000000 0.000000 1.982455 0.000000 1.805615
##    [57] 1.030816 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##    [65] 0.000000 2.141294 0.000000 2.758607 1.223213 3.336447 0.000000 0.000000
##    [73] 0.000000 0.000000 1.287397 0.000000 0.000000 0.000000 0.000000 0.000000
##    [81] 0.000000 1.539936 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##    [89] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##    [97] 0.000000 0.000000 0.000000 0.000000 1.238240 0.000000 0.000000 0.000000
##   [105] 0.000000 0.000000 0.000000 0.000000 0.000000 1.686106 2.638563 2.165261
##   [113] 0.000000 0.000000 0.000000 0.000000 0.000000 2.855774 0.000000 0.000000
##   [121] 0.000000 0.000000 0.000000 0.000000 1.636734 0.000000 2.293407 0.000000
##   [129] 0.000000 0.000000 0.000000 1.019504 0.000000 0.000000 0.000000 0.000000
##   [137] 0.000000 1.886250 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##   [145] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##   [153] 0.000000 0.000000 1.188038 4.507438 0.000000 0.000000 0.000000 0.000000
##   [161] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2.335511
##   [169] 0.000000 0.000000 1.035763 1.456043 0.000000 0.000000 0.000000 0.000000
##   [177] 0.000000 0.000000 1.008623 0.000000 0.000000 1.394117 0.000000 2.726671
##   [185] 0.000000 0.000000 0.000000 0.000000 0.000000 1.480907 0.000000 1.298985
##   [193] 0.000000 0.000000 2.117836 1.611030 0.000000 1.062991 0.000000 0.000000
##   [201] 1.107893 0.000000 0.000000 1.556534 3.315674 0.000000 0.000000 0.000000
##   [209] 0.000000 0.000000 0.000000 1.152697 1.197193 0.000000 0.000000 0.000000
##   [217] 1.538863 0.000000 0.000000 1.239855 0.000000 0.000000 0.000000 1.175953
##   [225] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##   [233] 0.000000 0.000000 0.000000 0.000000 1.298380 0.000000 0.000000 0.000000
##   [241] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.341415 1.219427
##   [249] 0.000000 2.169888 1.370921 0.000000 0.000000 0.000000 1.327956 2.936911
##   [257] 0.000000 0.000000 1.128278 1.238929 0.000000 0.000000 1.154930 1.100721
##   [265] 0.000000 1.046216 0.000000 1.486877 0.000000 0.000000 3.114801 1.078226
##   [273] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
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##   [297] 0.000000 0.000000 0.000000 0.000000 1.807932 0.000000 0.000000 2.007303
##   [305] 2.051611 1.117317 0.000000 0.000000 1.430270 0.000000 0.000000 0.000000
##   [313] 0.000000 0.000000 2.532423 2.525951 1.605401 0.000000 0.000000 0.000000
##   [321] 1.796572 0.000000 0.000000 0.000000 0.000000 1.049492 0.000000 0.000000
##   [329] 1.644314 0.000000 0.000000 0.000000 2.679102 0.000000 0.000000 0.000000
##   [337] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.685131
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##   [433] 0.000000 2.068552 3.419301 0.000000 0.000000 0.000000 3.667274 0.000000
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##   [457] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.930043
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##   [609] 2.407530 0.000000 0.000000 0.000000 1.165120 0.000000 4.142512 1.521510
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##   [665] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.022653
##   [673] 1.436899 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##   [681] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.490827
##   [689] 0.000000 0.000000 0.000000 0.000000 2.699338 0.000000 0.000000 0.000000
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##   [721] 0.000000 0.000000 0.000000 0.000000 1.320792 0.000000 0.000000 0.000000
##   [729] 1.349456 0.000000 0.000000 0.000000 0.000000 1.288877 0.000000 0.000000
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##   [745] 0.000000 1.918326 0.000000 1.253129 0.000000 0.000000 0.000000 2.065225
##   [753] 3.420339 0.000000 2.501262 0.000000 1.604430 0.000000 0.000000 0.000000
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##   [769] 1.168538 0.000000 0.000000 1.573374 0.000000 0.000000 0.000000 1.751207
##   [777] 0.000000 1.350008 0.000000 0.000000 0.000000 0.000000 0.000000 1.593722
##   [785] 0.000000 0.000000 1.276317 0.000000 0.000000 0.000000 1.004240 0.000000
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##   [817] 3.063346 0.000000 0.000000 2.278675 1.020061 3.281206 1.516409 0.000000
##   [825] 0.000000 1.789921 1.036869 1.590889 0.000000 0.000000 0.000000 2.424856
##   [833] 0.000000 0.000000 0.000000 1.922846 0.000000 2.135725 0.000000 0.000000
##   [841] 0.000000 0.000000 0.000000 1.810822 0.000000 0.000000 0.000000 0.000000
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##   [857] 0.000000 1.054571 0.000000 0.000000 0.000000 2.589648 0.000000 0.000000
##   [865] 0.000000 0.000000 0.000000 1.288839 0.000000 0.000000 0.000000 2.169950
##   [873] 1.354316 0.000000 0.000000 0.000000 1.885319 0.000000 0.000000 0.000000
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##   [913] 1.114131 1.206641 1.084699 0.000000 2.829504 1.848244 0.000000 0.000000
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##   [953] 0.000000 0.000000 0.000000 0.000000 2.357345 1.863119 0.000000 0.000000
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##   [969] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.777506 0.000000
##   [977] 0.000000 0.000000 2.662402 0.000000 0.000000 2.208211 1.716276 1.705594
##   [985] 0.000000 1.600239 1.397971 0.000000 0.000000 0.000000 0.000000 1.583876
##   [993] 0.000000 0.000000 0.000000 0.000000 1.150877 1.783388 0.000000 0.000000
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##  [9233] 0.000000 0.000000 1.467515 0.000000 1.320911 0.000000 0.000000 0.000000
##  [9241] 0.000000 1.111289 0.000000 2.263677 0.000000 0.000000 0.000000 0.000000
##  [9249] 0.000000 0.000000 0.000000 0.000000 0.000000 1.714958 0.000000 1.741142
##  [9257] 0.000000 0.000000 2.866635 0.000000 0.000000 1.303661 2.095727 0.000000
##  [9265] 0.000000 0.000000 0.000000 0.000000 0.000000 1.507356 0.000000 0.000000
##  [9273] 0.000000 0.000000 1.014189 0.000000 0.000000 0.000000 0.000000 1.249519
##  [9281] 0.000000 1.109832 0.000000 0.000000 1.447974 1.427167 0.000000 0.000000
##  [9289] 0.000000 0.000000 0.000000 1.977891 0.000000 0.000000 0.000000 1.410425
##  [9297] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9305] 0.000000 0.000000 1.530505 0.000000 0.000000 1.876965 0.000000 0.000000
##  [9313] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9321] 1.374455 0.000000 0.000000 0.000000 0.000000 2.666076 0.000000 0.000000
##  [9329] 0.000000 1.757414 0.000000 1.844230 0.000000 0.000000 0.000000 1.781859
##  [9337] 1.110611 0.000000 0.000000 0.000000 0.000000 0.000000 2.113687 0.000000
##  [9345] 0.000000 0.000000 0.000000 1.533539 0.000000 0.000000 0.000000 0.000000
##  [9353] 1.530215 1.338382 0.000000 3.723834 0.000000 1.509252 0.000000 0.000000
##  [9361] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9369] 0.000000 2.377407 0.000000 0.000000 0.000000 0.000000 0.000000 1.022279
##  [9377] 2.223875 3.640148 0.000000 0.000000 1.367317 0.000000 1.567369 0.000000
##  [9385] 0.000000 0.000000 1.337580 0.000000 1.738352 0.000000 0.000000 0.000000
##  [9393] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9401] 0.000000 0.000000 0.000000 0.000000 2.233868 0.000000 0.000000 0.000000
##  [9409] 0.000000 1.214969 0.000000 2.322683 0.000000 0.000000 2.343197 0.000000
##  [9417] 0.000000 0.000000 2.984804 0.000000 0.000000 0.000000 2.065855 0.000000
##  [9425] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9433] 0.000000 0.000000 1.587703 0.000000 1.009977 0.000000 0.000000 0.000000
##  [9441] 2.174472 0.000000 0.000000 0.000000 1.235113 0.000000 0.000000 0.000000
##  [9449] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9457] 0.000000 0.000000 0.000000 0.000000 2.247275 2.557698 1.469908 0.000000
##  [9465] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9473] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.128679 0.000000
##  [9481] 1.966326 0.000000 0.000000 0.000000 2.300120 0.000000 1.394367 0.000000
##  [9489] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9497] 0.000000 0.000000 1.317194 0.000000 0.000000 1.178518 0.000000 0.000000
##  [9505] 0.000000 0.000000 2.914969 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9513] 0.000000 2.000826 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9521] 0.000000 0.000000 0.000000 2.634761 0.000000 1.244068 0.000000 3.485452
##  [9529] 0.000000 1.609396 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9537] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9545] 0.000000 0.000000 2.373737 2.279727 0.000000 0.000000 0.000000 0.000000
##  [9553] 0.000000 0.000000 0.000000 0.000000 1.584295 0.000000 0.000000 0.000000
##  [9561] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9569] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2.602941
##  [9577] 1.058681 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9585] 1.999688 0.000000 0.000000 1.180759 0.000000 1.890856 0.000000 0.000000
##  [9593] 0.000000 1.408210 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9601] 0.000000 0.000000 0.000000 1.029606 0.000000 1.719958 0.000000 0.000000
##  [9609] 0.000000 0.000000 0.000000 0.000000 1.727255 0.000000 0.000000 1.134661
##  [9617] 0.000000 0.000000 0.000000 2.040330 0.000000 0.000000 1.111186 0.000000
##  [9625] 2.459436 0.000000 0.000000 0.000000 1.231233 0.000000 0.000000 2.935208
##  [9633] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9641] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9649] 0.000000 0.000000 0.000000 1.426504 0.000000 1.930832 0.000000 0.000000
##  [9657] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9665] 0.000000 0.000000 1.996591 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9673] 3.373716 0.000000 0.000000 2.875276 0.000000 1.571809 1.408576 0.000000
##  [9681] 1.722666 0.000000 0.000000 0.000000 1.733439 0.000000 0.000000 0.000000
##  [9689] 0.000000 1.306556 1.119723 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9697] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9705] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9713] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9721] 0.000000 0.000000 0.000000 0.000000 0.000000 1.565336 0.000000 0.000000
##  [9729] 0.000000 0.000000 1.637593 0.000000 0.000000 1.816051 1.116410 2.315409
##  [9737] 0.000000 0.000000 2.492586 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9745] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9753] 0.000000 1.553878 0.000000 0.000000 0.000000 1.147523 0.000000 2.528864
##  [9761] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.437007
##  [9769] 0.000000 0.000000 1.441058 0.000000 0.000000 0.000000 0.000000 2.169196
##  [9777] 2.873897 0.000000 1.100779 0.000000 3.425834 0.000000 0.000000 0.000000
##  [9785] 0.000000 0.000000 0.000000 0.000000 0.000000 1.588020 0.000000 0.000000
##  [9793] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9801] 0.000000 1.101056 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9809] 2.884000 0.000000 0.000000 1.728360 0.000000 1.838286 3.199950 0.000000
##  [9817] 0.000000 0.000000 0.000000 1.316825 1.038133 0.000000 0.000000 0.000000
##  [9825] 1.843623 1.113908 0.000000 0.000000 0.000000 0.000000 1.628693 0.000000
##  [9833] 0.000000 0.000000 1.323551 0.000000 1.140632 0.000000 0.000000 0.000000
##  [9841] 0.000000 0.000000 0.000000 0.000000 2.562239 3.049495 0.000000 0.000000
##  [9849] 0.000000 0.000000 0.000000 0.000000 0.000000 1.200871 0.000000 0.000000
##  [9857] 0.000000 0.000000 0.000000 0.000000 1.641557 0.000000 2.509780 0.000000
##  [9865] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2.205895
##  [9873] 0.000000 1.101037 1.872728 2.087784 0.000000 1.141848 0.000000 0.000000
##  [9881] 0.000000 1.351666 0.000000 0.000000 0.000000 1.699276 0.000000 0.000000
##  [9889] 1.655781 0.000000 1.087346 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9897] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9905] 0.000000 2.098967 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9913] 0.000000 0.000000 0.000000 1.955588 0.000000 1.314967 1.280683 0.000000
##  [9921] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9929] 0.000000 0.000000 0.000000 0.000000 0.000000 1.835759 1.141173 0.000000
##  [9937] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.153461 0.000000
##  [9945] 0.000000 1.990530 0.000000 1.267087 0.000000 0.000000 2.071997 0.000000
##  [9953] 0.000000 1.322367 1.192188 0.000000 1.617629 0.000000 0.000000 0.000000
##  [9961] 1.117882 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2.455912
##  [9969] 0.000000 0.000000 0.000000 0.000000 1.949525 1.565729 2.783529 0.000000
##  [9977] 0.000000 0.000000 0.000000 0.000000 5.369661 0.000000 0.000000 0.000000
##  [9985] 0.000000 3.697377 0.000000 0.000000 0.000000 2.070634 1.015189 1.619820
##  [9993] 1.223113 1.498826 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Ejercicio 1 (1.5 punto) - Inciso 2. continuación

EdeY = mean(Y)
VardeY =var(Y)

## [1] "El valor esperado de Yσ  es 0.422435072614802 y su varianza es 0.702419005981443"

Ejercicio 1 (1.5 punto) - Inciso 3.

3. Calcule el valor esperado de la media y de la varianza muestral de tamaño n = 10 de Yσ.

Y2 = muestras(n = 10, sigma = sqrt(2))

##  [1] 3.706362 0.000000 0.000000 0.000000 0.000000 1.221773 0.000000 1.512029
##  [9] 0.000000 0.000000

EdeY2 = mean(Y)
EVardeY2 = mean((Y - EdeY)^2)

## [1] "El valor esperado de Yσ2  es 0.422435072614802 y su varianza es 0.702348764080845"

Ejercicio 2 (1.0 punto) - Inciso 1.

Se desea hacer un intervalo de confianza de la media muestral: 1. Genere una muestra aleatoria de tamaño n = 10 con σ^2 = 2.

n = 10
sigma  = sqrt(2)

MuestraAleatoria = muestras(n, sigma)

##  [1] 0.000000 0.000000 1.513903 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 1.499087

Ejercicio 2 (1.0 punto) - Inciso 2.

Calcule el intervalo de confianza al 95% de la media muestral de esta muestra, suponiendo que se cumplen los supuestos de la distribución t-student y verifique si el intervalo contiene al valor real (tome como valor real, el valor esperado de la media muestral calculado previamente).

alpha = 0.05
gl =n-1

TdeSt = qt(alpha/2, df= gl, lower.tail = F)

ME = TdeSt*(sigma/sqrt(n))

Li <- mean(MuestraAleatoria) - ME
Ls <- mean(MuestraAleatoria) + ME

## [1] "Con un grado de confianza de 0.95 se estima que el valor real de la media poblacional se encuentra entre -0.710368462551572 y 1.31296641417037"

Se verifica si el intervalo contiene al valor real:

if(Li< EdeY & Ls > EdeY){
  print(paste('El valor esperado de Y se encuentra dentro del intervalo de confianza estimdo'))
  
} else {
  print(paste('El valor esperado de Y no se encuentra dentro del intervalo de confianza estimdo'))
}

## [1] "El valor esperado de Y se encuentra dentro del intervalo de confianza estimdo"

Ejercicio 2 (1.0 punto) - Inciso 3.

3. Repita este proceso con una simulación de Monte Carlo para estimar el verdadero grado de confianza del intervalo.

sigma= 2
n = 25
M = 100000

alpha = 0.05
gl = n - 1

t = qt(alpha/2, df= gl, lower.tail = F)

MEr = t*(sigma/sqrt(n))

xbar=rep(NA,M)
flag=rep(NA,M)

count=0

for (i in 1:M){
  x=rt(n, gl) 
  xbar[i]=mean(x)    
  LI=xbar[i]-MEr      
  LS=xbar[i]+MEr      
  
  if(LI < EdeY & LS > EdeY)
    flag[i]=1
  else 
    flag[i]=0
}

VGC = sum(flag)/M 

el verdadero grado de confianza del intervalo es:

## [1] 0.97286

Ejercicio 3 (2.5 punto) - Inciso 1 & 2.

Se desea verificar empıricamente el ejercicio hecho en clase.

1. En un bucle generar 10000 muestras aleatorias de tamaño n = 100 de una variable uniforme con parámetro 8.

2. Calcular θ1, θ2, θ3 para cada muestra en 3 vectores que llamará theta1, theta2 y theta3.

n = 100
M = 10000
theta = 8

theta1 = rep(NA,M)
theta2 = rep(NA,M)
theta3 = rep(NA,M)

for (i in 1:M){
  sim <- runif(n, min= 0 , max=theta)
  theta1[i] = 2 * mean(sim)
  theta2[i] = max(sim)
  theta3[i] = (n+1)*min(sim)
}

Ejercicio 3 (2.5 punto) - Inciso 3.

3. Dibuje los histogramas de los 3 estimadores. Utilice par(mfrow=c(1,3)) para que queden los 3 en la misma figura.

Ejercicio 3 (2.5 punto) - Inciso 4 & 5.

4. Unir los 3 vectores con los estimadores en una matriz. Utilice tidyverse.

matriz <- tibble(theta1, theta2, theta3)

5. Estimar el sesgo y la varianza para los 3 estimadores. Compare con los valores teóricos calculados en clase.

Edeθ1 <- mean(theta1)
Edeθ2 <- mean(theta2)
Edeθ3 <- mean(theta3)

sesgo1 <- Edeθ1 - theta
sesgo2 <- Edeθ2 - theta 
sesgo3 <- Edeθ3 - theta 

## [1] "Esperanza de Theta1 8.00030873473391 - Esperanza de 2 7.91994410190582 - Esperanza de Theta3 7.97669199499059"

## [1] "Sesgo de Theta1 0.000308734733909688 - Sesgo de 2 -0.0800558980941775 - Sesgo de Theta3 -0.0233080050094054"

Ejercicio 3 (2.5 punto) - Inciso 5. continuación

Vardeθ1 <- var(theta1) 
var1 <- theta^2/(3*n)

Vardeθ2 <- var(theta2)
var2 <- (n*theta^2)/((n+1)*(n+2)^2)

Vardeθ3 <- var(theta3)
var3 <- (n*theta^2)/(n+2)

## [1] "La varianza empírica es 0.21655747632024 y la varianza teórica 0.213333333333333"

## [1] "La varianza empírica es 0.00650517607892878 y la varianza teórica 0.00609057445536941"

## [1] "La varianza empírica es 64.4229278969176 y la varianza teórica 62.7450980392157"

Ejercicio 3 (2.5 punto) - Inciso 6.

6. Haga una gráfica de cajas y bigotes de la matriz, que incluya los 3 estimadores y trace una línea horizontal mostrando el valor real del parámetro θ.

Ejercicio 3 (2.5 punto) - Inciso 7.

7. Concluya y agregue comentarios que muestren su entendimiento sobre los estimadores.

De acuerdo con los hallado, podemos concluir que los estimadores theta1 es un estimador insesgado, pues, empirircamente se verifica que la diferencia entre la esperanza de este y el parámetro es muy cercana a 0. Sin embargo, el estimador más eficientes es theta2, pues es el que tiene menor varianza.