ACITIVIDAD 2

Problema1: Estimación del valor \(\pi\)

se realiza una funcion donde se calulara el valor de \(\pi\)

p1<- function(n) {

    x<-runif(n,0,1)
    y<-runif(n,0,1)

    formula<-(x-0.5)^2+ (y-0.5)^2
    
    prop<-c()

    for (i in 1:length(formula)) {

    if(formula[i] < 0.25) {
    prop[i]=1
  

    }

    if(formula[i] >= 0.25) {
    prop[i]=0
    
    }  
  
  
    }
    suma_1<-sum (prop)
    tetha<-suma_1/length(formula)
    salida<-list(repeticiones=n,
             cantidad_de_puntosque_cumplen_con_la_condicion=suma_1,
             estimador=tetha)
    return(salida)

    }
p1(1000)
## $repeticiones
## [1] 1000
## 
## $cantidad_de_puntosque_cumplen_con_la_condicion
## [1] 778
## 
## $estimador
## [1] 0.778
# 0.777

p1(10000)
## $repeticiones
## [1] 10000
## 
## $cantidad_de_puntosque_cumplen_con_la_condicion
## [1] 7869
## 
## $estimador
## [1] 0.7869
#0.7855


p1(100000)
## $repeticiones
## [1] 1e+05
## 
## $cantidad_de_puntosque_cumplen_con_la_condicion
## [1] 78574
## 
## $estimador
## [1] 0.78574
# 0.7854104
n Cantidad de ptos dentro del circulo π
1000 792 0.792
10000 7831 0.7831
100000 78645 0.7864

Problema 2: Propiedades de los estimadores

se utlizara una exponenecia de parametro es decir que el valor esperado sera =1

p2<- function(n) {
muestra<-data.frame()



for (i in 1:n) {#####el 10 l0 variamoes
  
  muestra[i,1:4]=expo<-rexp(4,rate =1)
  
  
}
muestra
colnames(muestra)<-c("X1","X2","X3","X4")
muestra
#n<-seq(1:10)
theta_1<-c()
theta_2<-c()
theta_3<-c()
theta_4<-c()
for (i in 1:n) {
 
theta_1[i]=((muestra[i,1]+muestra[i,2])/6)+((muestra[i,3]+muestra[i,4])/3)
theta_2[i]=(muestra[i,1]+(2*muestra[i,2])+(3*muestra[i,3])+(4*muestra[i,4]))/5  
theta_3[i]<-(muestra[i,1]+muestra[i,2]+muestra[i,3]+muestra[i,4])/4
theta_4[i]<-(min(muestra[i,1],muestra[i,2],muestra[i,3],muestra[i,4])+max(muestra[i,1],muestra[i,2],muestra[i,3],muestra[i,4]))/2



}


final_estimadores<-data.frame(muestra,theta_1,theta_2,theta_3,theta_4)
#final_estimadores


###EFICIENCIA
grafica1=boxplot(final_estimadores$theta_1,final_estimadores$theta_2,final_estimadores$theta_3,final_estimadores$theta_4,xlab = "ESTIMADORES THETA", whisklty = 2
                 ,col = c("blue", "yellow", "red","white"),names = c("theta_1","theta_2","theta_3","theta_4"))

######INSESGADEs
media_theta_1<-mean(theta_1)
media_theta_2<-mean(theta_2)
media_theta_3<-mean(theta_3)
media_theta_4<-mean(theta_4)

#eficiencia

varianza_theta_1<-var(theta_1)
varianza_theta_2<-var(theta_2)
varianza_theta_3<-var(theta_3)
varianza_theta_4<-var(theta_4)

salida<-list( media=c(media_theta_1,media_theta_2,media_theta_3,media_theta_4),
            varianza=c(varianza_theta_1,varianza_theta_2,varianza_theta_3,varianza_theta_4))
return(salida)


}

Cuando se tiene 20 repeticiones

Estimador 1 Estimador 2 estiamdor 3 estimador 4
Media 0.90 1.80 0.94 1.11
Varianza 0.19 0.72 0.15 0.21

Cuando se tiene 50 repeticiones

Estimador 1 Estimador 2 estiamdor 3 estimador 4
Media 0.84 1.64 0.86 1
Varianza 0.16 0.60 0.17 0.30

Cuando se tiene 100 repeticiones

Estimador 1 Estimador 2 estiamdor 3 estimador 4
Media 1 2 1 1.2
Varianza 0.34 1.59 0.27 0.46

Cuando se tiene 1000 repeticiones

Estimador 1 Estimador 2 estiamdor 3 estimador 4
Media 1 2 1 1.2
Varianza 0.28 1.24 0.24 0.39

Despues de Realizar las respectivas simulaciones con diferentes repeticiones se concluye que:

Problema 3: Teorema del limite central

Para este ejercicio se hace la sigueinte funcion:

ademas se desarrolla el punto c

p3<- function(n) {
Teorema<-matrix(,n,n)
exito<-rep(1,500)
fracaso<-rep(0,500)
lote<-c(exito,fracaso)

for (i in 1:n) {
  Teorema[i,]=sample(lote, n, replace = FALSE, prob = NULL) 
  
}
pi<-rowSums(Teorema)/n
promedio<-(mean(pi))
#print(pi)
#print(pi)
print(shapiro.test(pi))
print(qqnorm(pi))
print(promedio)

}
p3(100)
## 
##  Shapiro-Wilk normality test
## 
## data:  pi
## W = 0.98307, p-value = 0.2288

## $x
##   [1]  0.34512553  1.01522203  0.16365849  1.05812162 -1.20035886 -0.26631061
##   [7]  0.37185609  1.10306256  0.72247905 -0.08784484 -1.81191067  1.15034938
##  [13]  0.18911843  0.93458929  1.81191067 -1.37220381  0.53883603 -1.95996398
##  [19] -0.24042603  1.37220381 -1.15034938  0.21470157  1.20035886  0.97411388
##  [25] -0.93458929  0.24042603 -0.06270678 -0.56805150 -0.89647336  0.75541503
##  [31] -0.85961736 -0.21470157  0.39885507 -0.18911843 -0.03760829 -1.31057911
##  [37] -0.53883603 -0.01253347 -1.69539771  0.01253347  1.43953147  0.42614801
##  [43] -1.25356544 -1.10306256  1.25356544  0.78919165  2.17009038  0.26631061
##  [49]  0.82389363  1.31057911 -0.82389363 -1.59819314  2.57582930 -0.51007346
##  [55] -0.78919165 -0.75541503  0.56805150 -0.48172685 -0.72247905 -2.57582930
##  [61] -1.51410189  0.45376219 -1.43953147 -1.05812162  0.29237490 -0.16365849
##  [67]  0.03760829 -0.69030882 -2.17009038 -0.45376219  0.06270678 -0.42614801
##  [73] -1.01522203  1.51410189  0.08784484  0.11303854 -0.39885507  0.59776013
##  [79]  0.13830421 -0.65883769  0.85961736 -0.62800601  0.62800601 -0.37185609
##  [85]  0.48172685  1.59819314  0.31863936 -0.34512553 -0.31863936  1.69539771
##  [91]  0.65883769 -0.29237490 -0.13830421 -0.97411388  0.69030882 -0.59776013
##  [97]  0.89647336  0.51007346  1.95996398 -0.11303854
## 
## $y
##   [1] 0.52 0.56 0.51 0.56 0.46 0.49 0.52 0.56 0.54 0.50 0.42 0.57 0.51 0.55 0.59
##  [16] 0.43 0.53 0.40 0.49 0.58 0.46 0.51 0.57 0.55 0.47 0.51 0.50 0.48 0.47 0.54
##  [31] 0.47 0.49 0.52 0.49 0.50 0.43 0.48 0.50 0.42 0.50 0.58 0.52 0.45 0.46 0.57
##  [46] 0.54 0.60 0.51 0.54 0.57 0.47 0.42 0.63 0.48 0.47 0.47 0.53 0.48 0.47 0.39
##  [61] 0.42 0.52 0.42 0.46 0.51 0.49 0.50 0.47 0.39 0.48 0.50 0.48 0.46 0.58 0.50
##  [76] 0.50 0.48 0.53 0.50 0.47 0.54 0.47 0.53 0.48 0.52 0.58 0.51 0.48 0.48 0.58
##  [91] 0.53 0.48 0.49 0.46 0.53 0.47 0.54 0.52 0.59 0.49
## 
## [1] 0.5023

Por lo anterior, se tiene que el estimador tiende a el parametro poblacional ya que la cantidad de obervaciones, este es la regla de oro del teorema del limite centrar ya que por esto entre mas grande la muestra los estimadores seran muy cercanos a los parametros poblacionales.

Ademas por viendo la normalidad de los datos se ve que se cumple con este supuesto ya que el p valor es mayor al 5%.

punto D

p3(5)
## 
##  Shapiro-Wilk normality test
## 
## data:  pi
## W = 0.96086, p-value = 0.814

## $x
## [1] -0.4972006  0.0000000 -1.1797611  1.1797611  0.4972006
## 
## $y
## [1] 0.2 0.2 0.0 0.6 0.4
## 
## [1] 0.28
p3(10)
## 
##  Shapiro-Wilk normality test
## 
## data:  pi
## W = 0.89221, p-value = 0.1796

## $x
##  [1] -0.3754618  1.0004905 -1.5466353 -0.1225808 -0.6554235 -1.0004905
##  [7]  0.1225808  0.3754618  1.5466353  0.6554235
## 
## $y
##  [1] 0.5 0.8 0.3 0.5 0.4 0.3 0.5 0.5 0.9 0.6
## 
## [1] 0.53
p3(15)
## 
##  Shapiro-Wilk normality test
## 
## data:  pi
## W = 0.88864, p-value = 0.06395

## $x
##  [1]  0.7279133 -0.3406948 -0.7279133 -0.1678940 -1.2815516 -1.8339146
##  [7] -0.9674216  0.0000000  0.9674216  1.2815516  1.8339146  0.5244005
## [13]  0.1678940 -0.5244005  0.3406948
## 
## $y
##  [1] 0.6666667 0.5333333 0.4666667 0.5333333 0.3333333 0.2666667 0.3333333
##  [8] 0.5333333 0.6666667 0.6666667 0.6666667 0.6000000 0.5333333 0.4666667
## [15] 0.5333333
## 
## [1] 0.52
p3(20)
## 
##  Shapiro-Wilk normality test
## 
## data:  pi
## W = 0.98001, p-value = 0.9343

## $x
##  [1] -0.45376219  0.31863936  0.75541503 -1.15034938 -0.93458929 -0.06270678
##  [7]  0.45376219  1.95996398 -1.95996398  0.59776013 -0.75541503  1.15034938
## [13]  0.93458929 -0.59776013 -1.43953147  0.06270678  0.18911843  1.43953147
## [19] -0.31863936 -0.18911843
## 
## $y
##  [1] 0.45 0.55 0.60 0.40 0.40 0.50 0.55 0.75 0.25 0.55 0.40 0.65 0.60 0.40 0.30
## [16] 0.50 0.50 0.65 0.45 0.45
## 
## [1] 0.495
p3(30)
## 
##  Shapiro-Wilk normality test
## 
## data:  pi
## W = 0.9314, p-value = 0.05349

## $x
##  [1] -2.1280452 -0.7835004  0.0417893  0.1256613  1.3829941 -1.3829941
##  [7]  1.6448536  0.2104284 -0.2967378  0.2967378  1.1918162  0.3853205
## [13]  0.4770404  0.5729675  0.6744898 -1.1918162 -1.0364334  0.7835004
## [19] -0.6744898 -0.9027348  0.9027348  1.0364334 -0.5729675 -0.2104284
## [25] -0.1256613  2.1280452 -1.6448536 -0.4770404 -0.3853205 -0.0417893
## 
## $y
##  [1] 0.2666667 0.4333333 0.5000000 0.5000000 0.5666667 0.3666667 0.5666667
##  [8] 0.5000000 0.4666667 0.5000000 0.5333333 0.5000000 0.5000000 0.5000000
## [15] 0.5000000 0.3666667 0.3666667 0.5000000 0.4333333 0.3666667 0.5000000
## [22] 0.5000000 0.4333333 0.4666667 0.4666667 0.6666667 0.3333333 0.4333333
## [29] 0.4333333 0.4666667
## 
## [1] 0.4644444
p3(50)
## 
##  Shapiro-Wilk normality test
## 
## data:  pi
## W = 0.98618, p-value = 0.8211

## $x
##  [1]  1.88079361  0.27931903 -0.27931903 -1.22652812 -0.22754498  2.32634787
##  [7] -1.64485363 -0.17637416 -0.55338472 -0.12566135  1.47579103  1.22652812
## [13] -2.32634787 -0.49585035 -0.80642125  0.80642125  0.33185335 -1.47579103
## [19] -1.12639113 -1.34075503  1.34075503 -0.43991317 -0.73884685  0.55338472
## [25]  0.61281299 -1.88079361 -0.67448975  0.87789630  0.95416525 -0.38532047
## [31]  0.67448975  0.07526986  1.03643339  0.38532047  0.73884685 -1.03643339
## [37]  0.12566135  1.12639113  0.43991317 -0.33185335 -0.07526986 -0.61281299
## [43] -0.87789630  0.17637416 -0.95416525 -0.02506891  1.64485363  0.02506891
## [49]  0.49585035  0.22754498
## 
## $y
##  [1] 0.68 0.52 0.48 0.40 0.48 0.68 0.36 0.48 0.46 0.48 0.62 0.58 0.28 0.46 0.44
## [16] 0.56 0.52 0.38 0.40 0.38 0.60 0.46 0.44 0.54 0.54 0.34 0.44 0.56 0.56 0.46
## [31] 0.54 0.50 0.56 0.52 0.54 0.40 0.50 0.56 0.52 0.46 0.48 0.44 0.42 0.50 0.40
## [46] 0.48 0.62 0.48 0.52 0.50
## 
## [1] 0.4904
p3(60)
## 
##  Shapiro-Wilk normality test
## 
## data:  pi
## W = 0.97327, p-value = 0.2103

## $x
##  [1]  1.56891963 -0.06270678 -0.40791874  0.93458929  1.00133130 -0.02089009
##  [7] -1.15034938 -1.07286134 -2.39397980  0.02089009 -0.59776013  0.64849218
## [13] -0.36291730  0.27497775 -0.31863936  0.70095142  0.31863936 -1.95996398
## [19] -1.43953147  0.06270678 -1.00133130  1.07286134  0.75541503  1.15034938
## [25]  0.81221780  1.95996398 -0.27497775 -0.93458929 -0.75541503 -1.73166440
## [31]  0.10463346  0.36291730  0.40791874  1.23544034  0.87177097  1.33056151
## [37]  0.14674496 -0.54852228 -0.87177097 -1.33056151  1.43953147 -1.56891963
## [43]  0.45376219 -0.50058011 -0.23183436 -1.23544034 -0.45376219 -0.81221780
## [49] -0.70095142  1.73166440  0.18911843  0.23183436  2.39397980  0.50058011
## [55] -0.64849218 -0.18911843 -0.14674496 -0.10463346  0.54852228  0.59776013
## 
## $y
##  [1] 0.6000000 0.5166667 0.5000000 0.5666667 0.5666667 0.5166667 0.4333333
##  [8] 0.4333333 0.3666667 0.5166667 0.4833333 0.5500000 0.5000000 0.5333333
## [15] 0.5000000 0.5500000 0.5333333 0.3833333 0.4166667 0.5166667 0.4500000
## [22] 0.5666667 0.5500000 0.5666667 0.5500000 0.6166667 0.5000000 0.4500000
## [29] 0.4666667 0.3833333 0.5166667 0.5333333 0.5333333 0.5666667 0.5500000
## [36] 0.5666667 0.5166667 0.4833333 0.4500000 0.4166667 0.5666667 0.3833333
## [43] 0.5333333 0.4833333 0.5000000 0.4166667 0.4833333 0.4500000 0.4666667
## [50] 0.6000000 0.5166667 0.5166667 0.6500000 0.5333333 0.4666667 0.5000000
## [57] 0.5000000 0.5000000 0.5333333 0.5333333
## 
## [1] 0.5058333
p3(100)
## 
##  Shapiro-Wilk normality test
## 
## data:  pi
## W = 0.98221, p-value = 0.1968

## $x
##   [1]  0.75541503 -0.16365849 -1.81191067  0.97411388  0.21470157 -0.62800601
##   [7] -0.45376219 -0.59776013 -0.01253347  1.01522203 -1.10306256 -0.34512553
##  [13] -1.59819314 -0.89647336  0.24042603  0.01253347 -0.13830421  0.03760829
##  [19]  0.26631061  0.06270678 -2.57582930 -2.17009038  0.29237490 -0.85961736
##  [25]  0.78919165 -0.31863936 -0.82389363  0.31863936 -0.78919165  0.82389363
##  [31]  0.51007346 -0.11303854 -1.51410189  1.43953147  0.85961736  0.89647336
##  [37]  1.51410189  0.34512553  1.05812162 -1.37220381 -0.08784484  1.95996398
##  [43] -0.29237490 -0.26631061  1.10306256  2.57582930  0.08784484 -0.42614801
##  [49]  1.59819314  0.93458929 -0.06270678 -0.39885507 -0.37185609 -0.75541503
##  [55] -1.25356544 -0.56805150 -0.24042603  1.31057911  0.11303854 -1.95996398
##  [61]  0.53883603  0.37185609 -1.31057911  0.56805150 -1.05812162 -0.72247905
##  [67]  0.39885507 -1.69539771  0.59776013 -1.01522203  0.62800601  1.37220381
##  [73]  0.42614801 -0.97411388 -1.43953147  1.69539771 -1.20035886  1.15034938
##  [79] -0.69030882 -0.21470157  1.20035886  0.65883769 -0.93458929  0.45376219
##  [85]  0.13830421 -0.53883603 -0.03760829 -0.65883769 -0.51007346  0.69030882
##  [91]  0.16365849  1.25356544  0.72247905  0.18911843 -1.15034938  0.48172685
##  [97] -0.48172685  2.17009038  1.81191067 -0.18911843
## 
## $y
##   [1] 0.54 0.50 0.41 0.55 0.52 0.47 0.48 0.47 0.51 0.55 0.45 0.49 0.42 0.46 0.52
##  [16] 0.51 0.50 0.51 0.52 0.51 0.38 0.38 0.52 0.46 0.54 0.49 0.46 0.52 0.46 0.54
##  [31] 0.53 0.50 0.42 0.57 0.54 0.54 0.58 0.52 0.55 0.43 0.50 0.59 0.49 0.49 0.55
##  [46] 0.68 0.51 0.48 0.58 0.54 0.50 0.48 0.48 0.46 0.44 0.47 0.49 0.56 0.51 0.39
##  [61] 0.53 0.52 0.43 0.53 0.45 0.46 0.52 0.41 0.53 0.45 0.53 0.56 0.52 0.45 0.42
##  [76] 0.58 0.44 0.55 0.46 0.49 0.55 0.53 0.45 0.52 0.51 0.47 0.50 0.46 0.47 0.53
##  [91] 0.51 0.55 0.53 0.51 0.44 0.52 0.47 0.61 0.58 0.49
## 
## [1] 0.5004
p3(200)
## 
##  Shapiro-Wilk normality test
## 
## data:  pi
## W = 0.99051, p-value = 0.2119

## $x
##   [1]  0.285840875 -2.432379059  1.047215930  1.722383890  0.780664237
##   [6]  0.531604424 -1.162579875 -2.807033768  0.682377942 -0.068986959
##  [11]  0.119347567  0.797776846  0.924934461  0.698283366  1.918876226
##  [16] -0.056429069  1.422090432  0.131980140 -1.388450197 -0.815126333
##  [21] -0.797776846 -0.043880072 -2.241402728 -0.031337982 -0.018800820
##  [26]  0.815126333  0.298921424  0.546095926  0.944332036 -0.984234960
##  [31] -0.964091607 -0.531604424  0.832724719 -0.944332036  0.312053322
##  [36] -0.517223714  0.144633812  1.069154627 -2.004654462  1.534120544
##  [41]  0.850584865 -0.312053322 -0.780664237 -0.924934461  0.560703032
##  [46] -0.298921424 -0.285840875  0.157310685 -1.138288582 -0.682377942
##  [51]  0.575430769 -1.918876226 -1.356311745  0.170012889 -0.502949184
##  [56] -0.666643306  0.325239256 -0.651072016 -0.272809053 -0.905878812
##  [61] -0.635657014  0.338481986 -0.006266612 -0.259823400  0.182742585
##  [66] -1.114651015  0.590284394  1.091620367 -0.246881415  0.006266612
##  [71]  0.964091607  0.018800820  2.108358399 -1.845258117  0.351784345
##  [76] -0.887146559  2.004654462 -0.233980651 -0.868720547  0.714367440
##  [81] -0.488776411 -1.780464342  0.195501964 -1.325516200 -1.091620367
##  [86] -1.494672250  0.031337982 -0.474701147  0.365149249 -0.221118713
##  [91]  1.213339622  0.378579699 -0.620391602  1.114651015  1.239933478
##  [96] -0.460719309  0.868720547 -1.457421739 -0.208293252 -0.446826965
## [101] -0.763777244  0.392078788  0.405649708 -0.195501964 -1.295928846
## [106]  1.267434417  1.576111974 -0.433020331  0.419295753  0.043880072
## [111]  0.605269415  1.295928846  0.730638483 -1.069154627  0.433020331
## [116] -0.605269415 -0.590284394 -0.575430769 -1.669592577  0.056429069
## [121] -0.182742585  2.807033768  0.068986959  0.620391602  0.635657014
## [126]  0.446826965 -0.170012889  0.208293252 -1.047215930 -0.157310685
## [131] -0.419295753  1.138288582  0.747105302 -0.405649708 -0.144633812
## [136] -1.025770021  0.081555738  0.887146559 -0.747105302 -1.621082251
## [141]  2.241402728 -1.213339622 -0.392078788 -1.187577263  0.221118713
## [146]  0.984234960 -0.730638483  1.004785806 -1.267434417  0.651072016
## [151]  1.457421739 -0.378579699  2.432379059  1.621082251 -1.722383890
## [156]  1.025770021 -0.714367440 -0.365149249 -0.131980140  0.666643306
## [161]  1.325516200 -1.239933478 -0.351784345 -1.576111974 -0.119347567
## [166]  1.780464342  0.094137414  0.460719309  0.233980651  1.356311745
## [171] -0.560703032 -1.422090432 -0.338481986 -1.534120544  0.474701147
## [176] -0.546095926  1.162579875 -0.106734011  0.905878812  0.488776411
## [181]  0.106734011  1.494672250  1.669592577 -0.094137414 -0.081555738
## [186]  1.388450197  0.502949184  0.763777244 -0.325239256  0.246881415
## [191]  0.259823400 -2.108358399 -0.850584865  0.517223714  1.845258117
## [196]  1.187577263 -1.004785806  0.272809053 -0.832724719 -0.698283366
## 
## $y
##   [1] 0.505 0.425 0.530 0.550 0.520 0.510 0.465 0.415 0.515 0.495 0.500 0.520
##  [13] 0.525 0.515 0.555 0.495 0.540 0.500 0.455 0.475 0.475 0.495 0.430 0.495
##  [25] 0.495 0.520 0.505 0.510 0.525 0.470 0.470 0.485 0.520 0.470 0.505 0.485
##  [37] 0.500 0.530 0.440 0.545 0.520 0.490 0.475 0.470 0.510 0.490 0.490 0.500
##  [49] 0.465 0.480 0.510 0.440 0.455 0.500 0.485 0.480 0.505 0.480 0.490 0.470
##  [61] 0.480 0.505 0.495 0.490 0.500 0.465 0.510 0.530 0.490 0.495 0.525 0.495
##  [73] 0.570 0.440 0.505 0.470 0.555 0.490 0.470 0.515 0.485 0.440 0.500 0.455
##  [85] 0.465 0.450 0.495 0.485 0.505 0.490 0.535 0.505 0.480 0.530 0.535 0.485
##  [97] 0.520 0.450 0.490 0.485 0.475 0.505 0.505 0.490 0.455 0.535 0.545 0.485
## [109] 0.505 0.495 0.510 0.535 0.515 0.465 0.505 0.480 0.480 0.480 0.445 0.495
## [121] 0.490 0.600 0.495 0.510 0.510 0.505 0.490 0.500 0.465 0.490 0.485 0.530
## [133] 0.515 0.485 0.490 0.465 0.495 0.520 0.475 0.445 0.570 0.460 0.485 0.460
## [145] 0.500 0.525 0.475 0.525 0.455 0.510 0.540 0.485 0.575 0.545 0.440 0.525
## [157] 0.475 0.485 0.490 0.510 0.535 0.455 0.485 0.445 0.490 0.550 0.495 0.505
## [169] 0.500 0.535 0.480 0.450 0.485 0.445 0.505 0.480 0.530 0.490 0.520 0.505
## [181] 0.495 0.540 0.545 0.490 0.490 0.535 0.505 0.515 0.485 0.500 0.500 0.435
## [193] 0.470 0.505 0.550 0.530 0.465 0.500 0.470 0.475
## 
## [1] 0.49575
p3(500)
## 
##  Shapiro-Wilk normality test
## 
## data:  pi
## W = 0.99591, p-value = 0.2209

## $x
##   [1] -2.033520149  1.654627902 -3.090232306  0.603764838 -1.616436371
##   [6]  0.072756358 -0.683960672  2.226211769  0.729002718 -0.329205984
##  [11]  0.255936332 -0.597760126 -0.087844838  0.382622075  0.504371986
##  [16] -2.747781385 -0.323918153  1.674664889 -0.082813292  0.845198535
##  [21]  0.077783842  1.695397710  0.261119960  0.510073457  0.942376333
##  [26] -0.318639364 -0.077783842 -0.194224628  0.609791399  1.150349380
##  [31]  0.388021666  0.266310613  0.271508452 -0.189118426 -0.184017151
##  [36] -0.178920660 -1.121676528 -1.112321367  0.276713637 -0.874217165
##  [41]  1.716886018  1.926836573  0.515791557  1.160119883 -0.487364565
##  [46]  0.082813292  0.281926330  0.287146694  1.385171608  0.735557557
##  [51]  0.087844838  1.546433122 -0.481726850  2.033520149  0.092878609
##  [56]  0.615840189  2.074854734  0.852385798 -0.173828813  0.097914734
##  [61] -1.866295743  1.398376621 -1.103062556 -0.789191653 -1.346938626
##  [66] -0.072756358 -0.313369439 -0.168741468 -0.308108202  1.959963985
##  [71]  1.040731886  0.292374896  0.102953344  0.621911596  0.521526572
##  [76] -1.093897353 -0.067730713  0.859617364  0.297611102  0.302855481
##  [81] -0.677639965 -0.998576271 -0.062706778  1.049387085  0.950220942
##  [86] -1.695397710 -1.200358858  0.107994569  0.527278791 -0.476104403
##  [91] -0.782365165 -1.838423669 -0.470496968  0.113038541  1.275874179
##  [96]  0.118085389  0.866894167  0.958124465 -0.302855481  1.287270563
## [101]  0.308108202 -0.866894167  0.742144154 -0.591776891  0.628006014
## [106] -1.995393310  0.748763107 -0.671346215 -0.297611102 -0.057684425
## [111] -0.163658486  1.058121618 -0.158579730 -0.859617364  1.298836633
## [116] -1.334622287  0.313369439  1.786613365  1.310579112 -0.052663527
## [121]  0.123135248  0.318639364 -0.585814766 -0.665078946  1.066937632
## [126]  0.323918153  0.634123849 -0.153505060 -1.190118042 -0.775574943
## [131] -1.322505137 -0.148434341 -0.579873392  2.120071690  1.075837361
## [136]  1.811910673 -1.598193140 -0.464904288 -0.459326111 -0.990356294
## [141]  0.128188248 -0.453762190 -0.982202695  0.533048511  0.538836030
## [146] -0.768820293 -0.143367435 -0.292374896 -1.453806359 -0.658837693
## [151] -0.047643956 -2.120071690 -1.580466818 -0.573952419 -0.974113877
## [156]  0.393432594  2.365618127  0.640265509  0.966088297  1.411830078
## [161]  1.425544037  0.398855066  1.563223647 -1.674664889 -0.287146694
## [166] -0.652621998  0.646431416 -0.448212281 -0.568051498 -0.138304208
## [171]  1.170002408  2.457263390  0.874217165 -0.852385798  1.180000540
## [176] -0.845198535  0.881587347  1.322505137 -0.442676144 -1.310579112
## [181] -1.298836633 -1.084823128  2.575829304 -1.563223647  3.090232306
## [186] -0.838054670 -0.133244524 -0.646431416 -1.287270563 -0.640265509
## [191]  1.334622287 -1.180000540 -1.275874179 -2.074854734 -0.437153541
## [196] -1.546433122 -0.042625585  0.652621998  1.439531471 -1.530067588
## [201] -0.762100541 -0.128188248  0.133244524  0.544641655  1.084823128
## [206] -0.037608288 -1.075837361  0.658837693  0.404289290 -1.514101888
## [211]  0.974113877 -1.439531471 -0.830953321 -0.823893630 -1.425544037
## [216] -0.634123849 -0.431644239  0.409735480 -1.411830078 -0.562170292
## [221] -0.426148008  1.453806359 -0.420664620 -0.755415026  1.838423669
## [226]  0.138304208  0.415193851  0.143367435 -0.966088297  1.346938626
## [231] -0.958124465  0.420664620  0.755415026  0.665078946  0.762100541
## [236] -0.628006014 -0.556308467  1.580466818  0.148434341  0.426148008
## [241] -0.550465695 -0.281926330 -0.748763107  0.671346215  0.153505060
## [246]  0.329205984 -0.415193851  0.768820293 -2.290367878  0.334503036
## [251]  0.775574943 -1.959963985 -0.123135248  1.359462745 -0.544641655
## [256] -1.398376621 -0.032591937 -0.409735480 -0.027576406 -0.276713637
## [261] -0.404289290  1.995393310  0.339809491 -1.811910673  1.372203809
## [266] -1.170002408 -0.950220942 -0.538836030 -0.816874766  1.468383798
## [271] -0.942376333  1.190118042 -0.398855066  0.550465695 -1.066937632
## [276]  0.431644239 -0.022561568  1.093897353  2.747781385 -0.017547298
## [281] -1.058121618 -2.365618127 -0.934589291  0.677639965  0.556308467
## [286]  0.562170292 -0.012533470  0.683960672  0.889005731  0.896473364
## [291]  0.158579730 -1.049387085 -0.007519956  0.982202695 -1.786613365
## [296] -0.271508452 -1.385171608  0.782365165 -2.226211769  0.437153541
## [301] -0.393432594  0.903991328  1.483280127 -0.266310613 -1.040731886
## [306] -0.002506631  0.163658486  0.002506631  0.168741468  1.103062556
## [311] -0.742144154 -0.735557557 -1.654627902 -0.926858513  1.739197665
## [316] -0.729002718 -0.388021666 -0.118085389  0.442676144 -0.382622075
## [321] -1.160119883 -0.261119960 -1.264641136 -0.919182735 -1.762410298
## [326]  0.007519956  0.789191653 -0.533048511  0.990356294  1.200358858
## [331] -1.253565438  0.173828813 -0.255936332 -1.032153958  0.178920660
## [336]  1.112321367  1.866295743 -0.809895915  0.911560735 -0.722479052
## [341]  1.210727133 -0.527278791  1.498513068 -0.250759572  1.514101888
## [346]  0.184017151  2.170090378  0.448212281 -1.926836573 -0.245589523
## [351]  0.012533470 -0.621911596 -1.023651312 -0.113038541  1.121676528
## [356]  0.017547298 -0.615840189  0.345125531  0.350451343  0.568051498
## [361] -0.107994569  0.022561568  0.027576406 -0.521526572  0.453762190
## [366]  0.032591937 -0.515791557  1.221227222 -0.802956288  1.598193140
## [371]  0.189118426 -0.715985990 -0.377233617  1.131130901  0.037608288
## [376] -0.240426031  0.042625585  0.194224628  0.796055117 -2.575829304
## [381] -1.242641419 -1.498513068 -0.510073457  1.616436371  0.802956288
## [386]  0.998576271  0.690308824 -1.483280127 -0.235268941  0.459326111
## [391]  0.047643956 -1.739197665 -1.015222033 -1.231863709 -0.609791399
## [396]  0.696684917 -1.221227222 -0.230118101 -0.371856089 -0.366489294
## [401]  0.052663527  0.199335898  0.464904288 -0.504371986  0.057684425
## [406] -1.150349380 -2.170090378 -1.210727133  0.573952419 -0.709522974
## [411] -0.703089460  0.204452382 -1.140687476 -0.224973358 -0.911560735
## [416]  1.231863709 -2.457263390  0.919182735  0.355787114  0.361133034
## [421]  0.470496968 -0.219834564 -0.361133034  0.366489294  0.209574223
## [426] -0.214701568  0.809895915  0.476104403 -1.895697924 -0.903991328
## [431]  0.214701568  0.219834564 -1.635234015  0.481726850  1.006864279
## [436]  0.224973358  0.926858513 -0.696684917  0.703089460  0.579873392
## [441]  1.015222033 -0.896473364  1.762410298  1.242641419  0.371856089
## [446]  0.585814766  0.377233617  0.230118101 -0.498686864 -0.355787114
## [451] -0.889005731 -0.102953344  0.062706778  1.253565438 -1.131130901
## [456] -0.097914734  0.709522974  1.530067588  1.895697924 -0.350451343
## [461] -0.345125531  0.591776891  0.235268941 -0.092878609 -1.006864279
## [466] -1.468383798  0.715985990  1.023651312  1.264641136 -1.372203809
## [471]  0.722479052 -0.796055117 -0.603764838 -1.359462745  0.816874766
## [476]  1.140687476  0.487364565 -0.209574223 -0.493017814  0.597760126
## [481]  1.032153958 -0.204452382  0.493017814  0.823893630  0.067730713
## [486] -0.690308824  2.290367878 -0.199335898 -0.339809491 -0.881587347
## [491]  0.934589291  0.830953321  1.635234015 -0.334503036  0.240426031
## [496] -1.716886018  0.498686864  0.245589523  0.838054670  0.250759572
## 
## $y
##   [1] 0.468 0.528 0.448 0.510 0.474 0.502 0.490 0.538 0.512 0.496 0.504 0.492
##  [13] 0.500 0.506 0.508 0.456 0.496 0.528 0.500 0.514 0.502 0.528 0.504 0.508
##  [25] 0.516 0.496 0.500 0.498 0.510 0.520 0.506 0.504 0.504 0.498 0.498 0.498
##  [37] 0.482 0.482 0.504 0.486 0.528 0.532 0.508 0.520 0.494 0.502 0.504 0.504
##  [49] 0.524 0.512 0.502 0.526 0.494 0.534 0.502 0.510 0.536 0.514 0.498 0.502
##  [61] 0.470 0.524 0.482 0.488 0.478 0.500 0.496 0.498 0.496 0.532 0.518 0.504
##  [73] 0.502 0.510 0.508 0.482 0.500 0.514 0.504 0.504 0.490 0.484 0.500 0.518
##  [85] 0.516 0.472 0.480 0.502 0.508 0.494 0.488 0.470 0.494 0.502 0.522 0.502
##  [97] 0.514 0.516 0.496 0.522 0.504 0.486 0.512 0.492 0.510 0.468 0.512 0.490
## [109] 0.496 0.500 0.498 0.518 0.498 0.486 0.522 0.478 0.504 0.530 0.522 0.500
## [121] 0.502 0.504 0.492 0.490 0.518 0.504 0.510 0.498 0.480 0.488 0.478 0.498
## [133] 0.492 0.536 0.518 0.530 0.474 0.494 0.494 0.484 0.502 0.494 0.484 0.508
## [145] 0.508 0.488 0.498 0.496 0.476 0.490 0.500 0.466 0.474 0.492 0.484 0.506
## [157] 0.540 0.510 0.516 0.524 0.524 0.506 0.526 0.472 0.496 0.490 0.510 0.494
## [169] 0.492 0.498 0.520 0.540 0.514 0.486 0.520 0.486 0.514 0.522 0.494 0.478
## [181] 0.478 0.482 0.540 0.474 0.542 0.486 0.498 0.490 0.478 0.490 0.522 0.480
## [193] 0.478 0.466 0.494 0.474 0.500 0.510 0.524 0.474 0.488 0.498 0.502 0.508
## [205] 0.518 0.500 0.482 0.510 0.506 0.474 0.516 0.476 0.486 0.486 0.476 0.490
## [217] 0.494 0.506 0.476 0.492 0.494 0.524 0.494 0.488 0.530 0.502 0.506 0.502
## [229] 0.484 0.522 0.484 0.506 0.512 0.510 0.512 0.490 0.492 0.526 0.502 0.506
## [241] 0.492 0.496 0.488 0.510 0.502 0.504 0.494 0.512 0.464 0.504 0.512 0.468
## [253] 0.498 0.522 0.492 0.476 0.500 0.494 0.500 0.496 0.494 0.532 0.504 0.470
## [265] 0.522 0.480 0.484 0.492 0.486 0.524 0.484 0.520 0.494 0.508 0.482 0.506
## [277] 0.500 0.518 0.540 0.500 0.482 0.462 0.484 0.510 0.508 0.508 0.500 0.510
## [289] 0.514 0.514 0.502 0.482 0.500 0.516 0.470 0.496 0.476 0.512 0.464 0.506
## [301] 0.494 0.514 0.524 0.496 0.482 0.500 0.502 0.500 0.502 0.518 0.488 0.488
## [313] 0.472 0.484 0.528 0.488 0.494 0.498 0.506 0.494 0.480 0.496 0.478 0.484
## [325] 0.470 0.500 0.512 0.492 0.516 0.520 0.478 0.502 0.496 0.482 0.502 0.518
## [337] 0.530 0.486 0.514 0.488 0.520 0.492 0.524 0.496 0.524 0.502 0.536 0.506
## [349] 0.468 0.496 0.500 0.490 0.482 0.498 0.518 0.500 0.490 0.504 0.504 0.508
## [361] 0.498 0.500 0.500 0.492 0.506 0.500 0.492 0.520 0.486 0.526 0.502 0.488
## [373] 0.494 0.518 0.500 0.496 0.500 0.502 0.512 0.458 0.478 0.474 0.492 0.526
## [385] 0.512 0.516 0.510 0.474 0.496 0.506 0.500 0.470 0.482 0.478 0.490 0.510
## [397] 0.478 0.496 0.494 0.494 0.500 0.502 0.506 0.492 0.500 0.480 0.464 0.478
## [409] 0.508 0.488 0.488 0.502 0.480 0.496 0.484 0.520 0.460 0.514 0.504 0.504
## [421] 0.506 0.496 0.494 0.504 0.502 0.496 0.512 0.506 0.468 0.484 0.502 0.502
## [433] 0.472 0.506 0.516 0.502 0.514 0.488 0.510 0.508 0.516 0.484 0.528 0.520
## [445] 0.504 0.508 0.504 0.502 0.492 0.494 0.484 0.498 0.500 0.520 0.480 0.498
## [457] 0.510 0.524 0.530 0.494 0.494 0.508 0.502 0.498 0.482 0.474 0.510 0.516
## [469] 0.520 0.476 0.510 0.486 0.490 0.476 0.512 0.518 0.506 0.496 0.492 0.508
## [481] 0.516 0.496 0.506 0.512 0.500 0.488 0.538 0.496 0.494 0.484 0.514 0.512
## [493] 0.526 0.494 0.502 0.470 0.506 0.502 0.512 0.502
## 
## [1] 0.49974

Problema 4: Estimacción boostrap

Para este Ejercicio se desarrolla la siguiente función

p4<- function(n1,k) {

boos<-c(7.69, 4.97, 4.56, 6.49, 4.34, 6.24 , 4.45)
sample(boos, n1, replace = TRUE, prob = NULL) 

muestra1<-data.frame()


#n1<-30
#k<-100
for (i in 1:k) {#####el 10 l0 variamoes
  
  muestra1[i,1:n1]=sample(boos, n1, replace = TRUE, prob = NULL) 
  
  
}


muestra_media<-apply(muestra1,MARGIN=1,FUN =mean)
percentiles<-quantile(muestra_media,probs = c(0.025,0.975))

m1<-percentiles
m2<-c(2*mean(muestra_media)-percentiles[2],2*mean(muestra_media)-percentiles[1])
salida<-data.frame(m1,m2)
colnames(salida)<-c("METODO_1","METODO_2")
histograma<-hist(muestra_media,las=1,main="histograma boostrap")
abline(v=m1,col="blue",lwd=3)
abline(v=m2,col="red",lwd=3)

salidas_total<-list(
                    HIS=histograma,
                    INTERVALOS=salida  
                      
                      )

print(salidas_total)
#print(histograma)


}
p4(50,1000)

## $HIS
## $breaks
##  [1] 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2
## 
## $counts
##  [1]   1   5  17  59 134 199 234 174 108  43  19   5   2
## 
## $density
##  [1] 0.01 0.05 0.17 0.59 1.34 1.99 2.34 1.74 1.08 0.43 0.19 0.05 0.02
## 
## $mids
##  [1] 4.95 5.05 5.15 5.25 5.35 5.45 5.55 5.65 5.75 5.85 5.95 6.05 6.15
## 
## $xname
## [1] "muestra_media"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## $INTERVALOS
##       METODO_1 METODO_2
## 2.5%   5.21408 5.177657
## 97.5%  5.90054 5.864117

para estos dos intervalos de confianza se puede concluir que el metodo 2 es el que tienen el menor rango ademas este tiene una combinacion de los percentiles y el promedio , por otra parte el metodo 1 solo tiene encuenta los percentiles del 2.5 y el 97.5 %