Proyecto C. Proba. 2 Primavera 2023

Andrés Guerrero Robles Andrés Padrón Quintana

#Librerías a utilizar:
library(ggplot2)
library(esquisse)
library(dplyr)

Attaching package: 'dplyr'

The following objects are masked from 'package:stats':

    filter, lag

The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union

library(patchwork)

1. Considere X1, X2, . . ., Xn variables independientes e idènticamente distribuidas todas con distribución exponencial con media 5. También considere Y1, Y2, . . ., Ym variables aleatorias independientes e idénticamente distribuidas todas con distribución exponencial con media 15 e independientes de los Xi´s. Defina la cantidad.

                   T = (1/n Suma de i = 1 -> n Xi) /
                       (1/m Suma de i = 1 -> m Yj)

1. Tome m = 20. Simule n = 10, 100, 1000, 1000 Xi´s y Yj`s y sugiera una posible distribución.

n<-10

simulaciones <- replicate(n, {
  1/n * sum(rexp(n, rate = 0.2)) / (1/20 * sum(rexp(20, rate = 1/20)))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad1 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#2F4F4F", size = 1) + ggtitle("Densidad media n = 10") +
  theme_minimal()

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.

n<-100
simulaciones <- replicate(n, {
  1/n * sum(rexp(n, rate = 0.2)) / (1/20 * sum(rexp(20, rate = 1/20)))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad2 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#1E90FF", size = 1) + ggtitle("Densidad media n = 100") +
  theme_minimal()


n<-1000
simulaciones <- replicate(n, {
  1/n * sum(rexp(n, rate = 0.2)) / (1/20 * sum(rexp(20, rate = 1/20)))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad3 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#90EE90", size = 1) + ggtitle("Densidad media n = 1000") +
  theme_minimal()


n<-10000
simulaciones <- replicate(n, {
  1/n * sum(rexp(n, rate = 0.2)) / (1/20 * sum(rexp(20, rate = 1/20)))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad4 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#D87093", size = 1) + ggtitle("Densidad media n = 10,000") +
  theme_minimal()

(densidad1 | densidad2) / (densidad3 | densidad4)

Parece una distribución normal con media aproximada en 0.25, la pecualiaridad es que tiene sesgo positivo, más simulaciones tienden a 0.

2. Tome n = 20. Simule n = 10, 100, 1000, 1000 Xi´s y Yj`s y sugiera una posible distribución.

n<-10

simulaciones <- replicate(n, {
  1/20* sum(rexp(20, rate = 0.2)) / (1/n * sum(rexp(n, rate = 1/20)))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad11 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#6A5ACD", size = 1) + ggtitle("Densidad media n = 10") +
  theme_minimal()

n<-100

simulaciones <- replicate(n, {
  1/20* sum(rexp(20, rate = 0.2)) / (1/n * sum(rexp(n, rate = 1/20)))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad12 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#008080", size = 1) + ggtitle("Densidad media n = 100") +
  theme_minimal()

n<-1000
simulaciones <- replicate(n, {
  1/20* sum(rexp(20, rate = 0.2)) / (1/n * sum(rexp(n, rate = 1/20)))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad13 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#4692B4", size = 1) + ggtitle("Densidad media n = 1000") +
  theme_minimal()

n<-10000
simulaciones <- replicate(n, {
  1/20* sum(rexp(20, rate = 0.2)) / (1/n * sum(rexp(n, rate = 1/20)))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad14 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#D8BFD8", size = 1) + ggtitle("Densidad media n = 100,000") +
  theme_minimal()

(densidad11 | densidad12) / (densidad13 | densidad14)

Parece una distribución normal con media aproximada en 0.25, la pecualiaridad es que tiene sesgo positivo, más simulaciones tienden a 0.

3. Encuentre la distribución de T Con las observaciones anteriores podemos asumir que T presenta una distribución normal, ya que sabemos que división de normales es Normal.

2. Considere X1, X2, . . ., Xn variables aleatorias independientes e idénticamente distribuidas todas con dsitribución Unif(0, 5). Defina:

                       U = (raíz de n)[X1 + X2 + . . . +Xn]/
                             X1^2 + X2^2 + . . . + Xn^2

Haga n = 10, 100, 1000, 100000 simulaciones de U y sugiera una posible distribución para U.

n<-10

simulaciones <- replicate(n, {
  sqrt(n)*sum(rnorm(n, mean = 0, sd = 1))/sum(rchisq(n, df=1))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad20 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#A0522D", size = 1) + ggtitle("Densidad media n = 10") +
  theme_minimal()

n<-100
simulaciones <- replicate(n, {
  sqrt(n)*sum(rnorm(n, mean = 0, sd = 1))/sum(rchisq(n, df=1))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad21 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#C0C0C0", size = 1) + ggtitle("Densidad media n = 100") +
  theme_minimal()


n<-1000
simulaciones <- replicate(n, {
  sqrt(n)*sum(rnorm(n, mean = 0, sd = 1))/sum(rchisq(n, df=1))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad22 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#FF6347", size = 1) + ggtitle("Densidad media n = 1000") +
  theme_minimal()

n<-10000
simulaciones <- replicate(n, {
  sqrt(n)*sum(rnorm(n, mean = 0, sd = 1))/sum(rchisq(n, df=1))
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad23 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#9ACD32", size = 1) + ggtitle("Densidad media n = 10,000") +
  theme_minimal()

(densidad20 | densidad21) / (densidad22 | densidad23)

La función U, con más simulaciones se va pareciendo au una Normal Estándar. N-(0,1).

3. Considere X1, X2, . . ., Xn variables aleatorias independientes e idénticamente distribuidas todas con dsitribución Unif(0, 5). Defina: U = máx{X1, X2, . . ., Xn}-5

Haga n = 10, 100, 1000, 100000 simulaciones de U y sugiera una posible distribución para U.

n<-10

simulaciones <- replicate(n, {
  max(runif(n,0,5)) - 5
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad31 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#40E0D0", size = 1) + ggtitle("Densidad media n = 10") +
  theme_minimal()

n<-100

simulaciones <- replicate(n, {
  max(runif(n,0,5)) - 5
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad32 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#FF5347", size = 1) + ggtitle("Densidad media n = 100") +
  theme_minimal()

n<-1000

simulaciones <- replicate(n, {
  max(runif(n,0,5)) - 5
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad33 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#A0522D", size = 1) + ggtitle("Densidad media n = 1000") +
  theme_minimal()

n<-10000

simulaciones <- replicate(n, {
  max(runif(n,0,5)) - 5
})

df_simuls <- data.frame(simulaciones = simulaciones)
densidad34 <- df_simuls %>% ggplot() +
  geom_density(aes(x = simulaciones), color = "#AFEEEE", size = 1) + ggtitle("Densidad media n = 10,000") +
  theme_minimal()

(densidad31 | densidad32) / (densidad33 | densidad34)

Parece que tiene forma de una distribución Normal con media negativa y sesgo negativo.

4. Se lanzan 3 dados balanceados. Sea X la suma de las caras.

1. Para n = 100,1000,10000,100000 haga n simulaciones para obtener la función de masa de probabilidad aproximada de X.

caras_del_dado <- 1:6
num_dados <- 3
num_sim <- 100000
vector_simuls <- c()
for(i in 1:num_sim){
  simulacion <- sample(x = caras_del_dado, size = num_dados,replace = TRUE)
  valor_simulacion <- sum(simulacion)
  vector_simuls <- c(vector_simuls,valor_simulacion)
}

str(vector_simuls)

 int [1:100000] 12 13 8 7 11 8 8 8 9 10 ...

d <- data.frame(suma = vector_simuls)
g100 <- ggplot(data.frame(suma=d[1:100,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#2461A3") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n = 100") +
    theme_minimal()
g1000 <- ggplot(data.frame(suma=d[1:1000,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#9B59B6") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n = 1000") +
    theme_minimal()
g10000 <- ggplot(data.frame(suma=d[1:10000,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#76D7C4") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n = 10000") +
    theme_minimal()
g100000 <- ggplot(data.frame(suma=d[1:100000,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#F1C40F") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n = 100000") +
    theme_minimal()

2. Usando las funciones de las librería ’patchwork‘ y ’ggplot2‘ ponga en un mismo gráfico las cuatro gráficas de las funciones de masa del inicio anterior.

(g100 + g1000) / (g10000 + g100000)

3. Para n = 10, . . . , 100000 haga n simulaciones para obtener un estimado de la probabilidad P(X ≤ 3). Ponga en un gráfico n en el eje horizontal y la probabilidad en el eje vertical.

simuls_satisfacen <- vector_simuls<=3
str(simuls_satisfacen)

 logi [1:100000] FALSE FALSE FALSE FALSE FALSE FALSE ...

#simuls_satisfacen es un vector de booleanos que cumplen la condición

probabilidades_estimadas <- c()
for(i in 10:100000){
  satisfacen <- simuls_satisfacen[1:i]
  proba_estimada <- sum(satisfacen)/length(satisfacen)
  probabilidades_estimadas <- c(probabilidades_estimadas, proba_estimada)
}

df_probas <- data.frame(n = 10:100000,probabilidades = probabilidades_estimadas)
df_probas %>% ggplot() +
  geom_line(aes(x = n, y = probabilidades)) +
  theme_light()

5. Una urna tiene 7 bolas numeradas del 1 al 7. Se sacan 2 bolas de la urna SIN reemplazo. Sea X la suma del nú́meros en las bolas.

1. Para n = 100,1000,10000,100000 haga n simulaciones para obtener la función de masa de probabilidad aproximada de X.

num_bola <- 1:7
bolas_selec <- 2
num_sim <- 100000
vector_simuls <- c()
for(i in 1:num_sim){
  simulacion <- sample(x = num_bola, size = bolas_selec,replace=FALSE)
  valor_simulacion <- sum(simulacion)
  vector_simuls <- c(vector_simuls,valor_simulacion)
}

str(vector_simuls)

 int [1:100000] 10 11 6 9 9 9 9 8 3 9 ...

d <- data.frame(suma = vector_simuls)
g100 <- ggplot(data.frame(suma = d[1:100,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#E74C3C") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n = 100") +
    theme_minimal()
g1000 <- ggplot(data.frame(suma = d[1:1000,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#27AE60") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n = 1000") +
    theme_minimal()
g10000 <- ggplot(data.frame(suma = d[1:10000,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#E832A9") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n = 10000") +
    theme_minimal()
g100000 <- ggplot(data.frame(suma = d[1:100000,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#F1F12F") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n = 100000") +
    theme_minimal()

2. Usando las funciones de las librería ’patchwork‘ y ’ggplot2‘ ponga en un mismo gráfico las cuatro gráficas de las funciones de masa del inicio anterior.

(g100 + g1000) / (g10000 + g100000)

3. Para n = 10, . . . , 100000 haga n simulaciones para obtener un estimado de la probabilidad P(X ≤ 10). Ponga en un gráfico n en el eje horizontal y la probabilidad en el eje vertical.

simuls_satisfacen <- vector_simuls<=10
str(simuls_satisfacen)

 logi [1:100000] TRUE FALSE TRUE TRUE TRUE TRUE ...

#simuls_satisfacen es un vector de booleanos que cumplen la condición

probabilidades_estimadas <- c()
for(i in 10:100000){
  satisfacen <- simuls_satisfacen[1:i]
  proba_estimada <- sum(satisfacen)/length(satisfacen)
  probabilidades_estimadas <- c(probabilidades_estimadas, proba_estimada)
}

df_probas <- data.frame(n = 10:100000,probabilidades=probabilidades_estimadas)
df_probas %>% ggplot() +
  geom_line(aes(x = n, y = probabilidades)) +
  theme_light()

6. Una urna tiene 7 bolas numeradas del 1 al 7. Se sacan 2 bolas de la urna CON reemplazo. Sea X la suma del nú́meros en las bolas.

1. Para n = 100,1000,10000,100000 haga n simulaciones para obtener la función de masa de probabilidad aproximada de X.

num_bola <- 1:7
bolas_selec <- 2
num_sim <- 100000
vector_simuls <- c()
for(i in 1:num_sim){
  simulacion <- sample(x = num_bola, size = bolas_selec,replace = TRUE)
  valor_simulacion <- sum(simulacion)
  vector_simuls <- c(vector_simuls,valor_simulacion)
}

str(vector_simuls)

 int [1:100000] 14 3 5 8 9 12 3 9 8 9 ...

d <- data.frame(suma = vector_simuls)
g100 <- ggplot(data.frame(suma = d[1:100,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#36B034") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n=100") +
    theme_minimal()
g1000 <- ggplot(data.frame(suma = d[1:1000,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#26ACAA") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n=1000") +
    theme_minimal()
g10000 <- ggplot(data.frame(suma = d[1:10000,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#7533AE") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n=10000") +
    theme_minimal()
g100000 <- ggplot(data.frame(suma = d[1:100000,])) +
    aes(x = suma) +
    geom_histogram(bins = 30L, fill = "#AC2142") +
    labs(x = "Suma", y = "f", title = "Función masa de la suma de n=100000") +
    theme_minimal()

2. Usando las funciones de las librería ’patchwork‘ y ’ggplot2‘ ponga en un mismo gráfico las cuatro gráficas de las funciones de masa del inicio anterior.

(g100 + g1000) / (g10000 + g100000)

3. Para n = 10, . . . , 100000 haga n simulaciones para obtener un estimado de la probabilidad P(X ≤ 10). Ponga en un gráfico n en el eje horizontal y la probabilidad en el eje vertical.

simuls_satisfacen <- vector_simuls<=10
str(simuls_satisfacen)

 logi [1:100000] FALSE TRUE TRUE TRUE TRUE FALSE ...

#simuls_satisfacen es un vector de booleanos que cumplen la condición

probabilidades_estimadas <- c()
for(i in 10:100000){
  satisfacen <- simuls_satisfacen[1:i]
  proba_estimada <- sum(satisfacen)/length(satisfacen)
  probabilidades_estimadas <- c(probabilidades_estimadas, proba_estimada)
}

df_probas <- data.frame(n=10:100000,probabilidades =probabilidades_estimadas)
df_probas %>% ggplot() +
  geom_line(aes(x = n, y = probabilidades)) +
  theme_light()

7. En un salón de clases del curso de Cálculo de Probabilidades 2 hay 50 estudiantes. Cada estudiante pone en papel su clave única (CU) y la introduce en una urna común. Posteriormente dichxs estudiantes toman un papel de la urna. Sea X el número de alumnos que sacaron el papelito con su clave única

1. Para n = 100,1000,10000,100000 haga n simulaciones para obtener la función de masa de probabilidad aproximada de X.

total <- c()
for(i in 1:100000){
  a <- sample.int(50,50,replace=FALSE)
  b <- sample.int(50,50,replace=FALSE)
  contador <- 0
  for(i in 1:50){
    if (a[i]==b[i]){
      contador <- contador+1
    }
  }
  total <- c(total,contador)
}
str(total)

 num [1:100000] 0 1 0 0 0 4 1 3 1 1 ...

`

d <- data.frame(match=total)
g100 <- ggplot(data.frame(match=d[1:100,])) +
    aes(x=match) +
    geom_histogram(bins = 30L, fill = "#D64B25") +
    labs(x = "Match", y = "f", title = "Función masa del match de n = 100") +
    theme_minimal()
g1000 <- ggplot(data.frame(match=d[1:1000,])) +
    aes(x=match) +
    geom_histogram(bins = 30L, fill = "#7BBC12") +
    labs(x = "Match", y = "f", title = "Función masa del match de n = 1000") +
    theme_minimal()
g10000 <- ggplot(data.frame(match=d[1:10000,])) +
    aes(x=match) +
    geom_histogram(bins = 30L, fill = "#CC1647") +
    labs(x = "Match", y = "f", title = "Función masa del match de n = 10000") +
    theme_minimal()
g100000 <- ggplot(data.frame(match=d[1:100000,])) +
    aes(x=match) +
    geom_histogram(bins = 30L, fill = "#5535DE") +
    labs(x = "Match", y = "f", title = "Función masa del match de n = 100000") +
    theme_minimal()

2. Usando las funciones de las librería ’patchwork‘ y ’ggplot2‘ ponga en un mismo gráfico las cuatro gráficas de las funciones de masa del inicio anterior.

(g100 + g1000) / (g10000 + g100000)

3. Para n = 10, . . . , 100000 haga n simulaciones para obtener un estimado de la probabilidad P(X ≤ 15). Ponga en un gráfico n en el eje horizontal y la probabilidad en el eje vertical.

simuls_satisfacen <- total<=15
str(simuls_satisfacen)

 logi [1:100000] TRUE TRUE TRUE TRUE TRUE TRUE ...

#simuls_satisfacen es un vector de booleanos que cumplen la condición

probabilidades_estimadas <- c()
for(i in 10:100000){
  satisfacen <- simuls_satisfacen[1:i]
  proba_estimada <- sum(satisfacen)/length(satisfacen)
  probabilidades_estimadas <- c(probabilidades_estimadas, proba_estimada)
}

df_probas <- data.frame(n=10:100000,probabilidades=probabilidades_estimadas)
df_probas %>% ggplot() +
  geom_line(aes(x = n, y = probabilidades)) +
  theme_light()

8. Simule tantos números uniformes en el intervalo (0,1) hasta que su suma sea mayor o igual que 1. Sea N el número de sumandos requeridos para alcanzar dicho objetivo. Por ejemplo si obtuvo los nùmeros 0.35, 0.58, 0.22 se tiene que N = 3 (pues se necesitó 3 sumandos para que la suma sea mayor ó igual que 1).

1. Para n = 100, 1000, 100000, 100000 haga n simulaciones para obtener la función de masa de probabilidad de N

sumaN <- function() {
  sum = 0
  N = 0
       while(sum < 1)
       {
       a = runif(1, 0, 1)
       sum = sum + a
       N = N + 1
       }
  return(N)
}

x <- replicate(100, sumaN())
df <- data.frame(suma = x)

grafica51 <- ggplot(df) +
  geom_histogram(aes(x = suma, y = ..count../sum(..count..)),
                 size = 1, fill = "#5F9EA0", bins = 30L) +
  ggtitle("F. masa de suma n = 100") + ylab("f") + xlab("suma") + 
  theme_light()

x2 <- replicate(1000, sumaN())
df <- data.frame(suma = x2)

grafica52 <- ggplot(df) +
  geom_histogram(aes(x = suma, y = ..count../sum(..count..)),
                 size = 1, fill = "#8B0000", bins = 30L) +
  ggtitle("F. masa de suma n = 1000") + ylab("f") + xlab("suma") + 
  theme_light()

x3 <- replicate(10000, sumaN())
df <- data.frame(suma = x3)

grafica53 <- ggplot(df) +
  geom_histogram(aes(x = suma, y = ..count../sum(..count..)),
                 size = 1, fill = "#20B2AA", bins = 30L) +
  ggtitle("F. masa de suma n = 10,000") + ylab("f") + xlab("suma") + 
  theme_light()

x4 <- replicate(100000, sumaN())
df <- data.frame(suma = x4)

grafica54 <- ggplot(df) +
  geom_histogram(aes(x = suma, y = ..count../sum(..count..)),
                 size = 1, fill = "#C71585", bins = 30L) +
  ggtitle("F. masa de suma n = 100,000") + ylab("f") + xlab("suma") + 
  theme_light()

2. Usando las funciones de la librería patchwork y ggplot2 ponga en un mismo gráfico las cuatro gráficas de las funciones de densidad del inicio a)

(grafica51 | grafica52) / (grafica53 | grafica54)

Warning: The dot-dot notation (`..count..`) was deprecated in ggplot2 3.4.0.
ℹ Please use `after_stat(count)` instead.

3. Para n = 10, . . ., 100, 000 haga n simulaciones para obtener un estimado del valor esperado de E(N). Ponga en un gráfico n en el eje horizontal y el valor esperado en el eje vertical.

simuls_satisfacen <- x4
i <- 10
satisfacen <- simuls_satisfacen[1:i]
proba_estimada <- sum(satisfacen)/length(satisfacen)

probabilidades_estimadas <- c()
for(i in 10:length(simuls_satisfacen)){
  satisfacen <- simuls_satisfacen[1:i]
                  proba_estimada <- sum(satisfacen)/length(satisfacen)
                  probabilidades_estimadas <- c(probabilidades_estimadas, proba_estimada)
}

df_probas <-data.frame(n = 10:length(simuls_satisfacen),
                       probabilidades = probabilidades_estimadas)

grafica8c <- ggplot(df_probas) +
geom_line(aes(x = n, y = probabilidades)) + ylab("Esperanza") + theme_light()

grafica8c

9. Considere X1, . . . , Xn variables aletorias independientes e idénticamente distribuidas, Xi-Unif(0,1). Sea X(2) el segundo valor mas pequeño de estas n Xi´s

1. Para n = 100,1000,10000,100000 haga 10,000 simulaciones para obtener la función de densidad de probabilidad aproximada de X(2).

total <- 100000
simulaciones <- 10000
vsimulaciones <- c()
respy <- c()
for(i in 1:simulaciones){
  respy <- c(runif(total, min = 0, max = 1))
  respy2 <- sort(respy, decreasing = FALSE)
  vsimulaciones <- c(vsimulaciones, respy2[2])
  
}

df <- data.frame(segundo = vsimulaciones)
densidad91 <- df %>% ggplot() +
  geom_density(aes(x = segundo), color = "#48D1CC", linewidth = 1) +
  ggtitle("Densidad 2do menor n = 100,000") +
  ylab("f") + xlab("Segundo menor") +
  theme_minimal()


total <- 100
vsimulaciones2 <- c()
respy <- c()
for(i in 1:simulaciones){
  respy <- c(runif(total, min = 0, max = 1))
  respy2 <- sort(respy, decreasing = FALSE)
  vsimulaciones2 <- c(vsimulaciones2, respy2[2])
  
}

df <- data.frame(segundo = vsimulaciones2)
densidad92 <- df %>% ggplot() +
  geom_density(aes(x = segundo), color = "#000080", linewidth = 1) +
  ggtitle("Densidad 2do menor n = 100") +
  ylab("f") + xlab("Segundo menor") +
  theme_minimal()


total <- 1000
vsimulaciones3 <- c()
respy <- c()
for(i in 1:simulaciones){
  respy <- c(runif(total, min = 0, max = 1))
  respy2 <- sort(respy, decreasing = FALSE)
  vsimulaciones3 <- c(vsimulaciones3, respy2[2])
  
}

df <- data.frame(segundo = vsimulaciones)
densidad93 <- df %>% ggplot() +
  geom_density(aes(x = segundo), color = "#D87093", linewidth = 1) +
  ggtitle("Densidad 2do menor n = 1000") +
  ylab("f") + xlab("Segundo menor") +
  theme_minimal()


total <- 10000
vsimulaciones4 <- c()
respy <- c()
for(i in 1:simulaciones){
  respy <- c(runif(total, min = 0, max = 1))
  respy2 <- sort(respy, decreasing = FALSE)
  vsimulaciones4 <- c(vsimulaciones4, respy2[2])
  
}

df <- data.frame(segundo = vsimulaciones4)
densidad94 <- df %>% ggplot() +
  geom_density(aes(x = segundo), color = "#BC8F8F", linewidth = 1) +
  ggtitle("Densidad 2do menor n = 10,000") +
  ylab("f") + xlab("Segundo menor") +
  theme_minimal()

2. Usando las funciones de la librería patchwork y ggplot2 ponga en un mismo gráfico las cuatro gráficas de las funciones de densidad del inicio a)

(densidad92 | densidad93) / (densidad94 | densidad91)

3. ¿Qué densidad diría que tiene X(2)?

Tiene forma de una distribución Normal con media casi 0, también tiene sesgo positivo.

4. ¿Es consistente este resultado con el ejemplo teórico visto en clase?

En clase se obtuvo que la j-ésima funcion se distribuía normal con parametros (j, n-j+1). Pedir la X(2) seria como pedir una beta con parametros (2, n-2+1) con n muy grande o que tiende a infinito. Esta distribucion Beta debería converger (por TLC) a una normal con media casi 0 que es justamente lo que se observa en las simulaciones obtenidas.

10. Considere X1, . . . , Xn variables aletorias independientes e idénticamente distribuidas, Xi-Unif(-1,1). Sea Xmed la mediana de estas n Xi´s, i.e. Xmed = X(n+1/2) si n es impar 1/2(X(n/2) + X(n/2+1)) si n es par

1. Para n = 100,1000,10000,100000 haga 10,000 simulaciones para obtener la función de densidad de probabilidad aproximada de Xmed.

n <- 100

medianaU <- replicate(100000, { 
  Xbarra <- median(runif(n,-1,1))
  Xbarra
})

df_simuls <- data.frame(medianaU = medianaU)

grafica10 <- ggplot(df_simuls) +
  geom_histogram(aes( x= medianaU,
                      y = ..density..), bins=30L, size = 1, fill = "#1E90FF", alpha = 0.5) +
  theme_minimal()

df_simuls

           medianaU
1      7.960894e-03
2     -3.324228e-02
3      1.042523e-01
4      6.076367e-02
5     -7.370296e-02
6      1.236822e-01
7     -8.673262e-02
8      4.226863e-02
9     -1.536381e-01
10    -5.983922e-02
11     4.985847e-02
12    -5.135634e-03
13    -5.352164e-02
14    -2.286588e-02
15    -1.762612e-01
16    -6.656073e-02
17    -5.661804e-02
18     7.220142e-03
19     1.154757e-01
20     1.394685e-01
21     3.612405e-02
22    -3.016869e-03
23    -1.136155e-02
24     3.184528e-01
25    -4.810926e-03
26    -2.108659e-01
27     5.970824e-02
28    -2.036199e-01
29    -1.346008e-01
30     2.954334e-02
31     1.777308e-01
32     7.330171e-02
33    -6.671728e-02
34    -1.715374e-01
35    -1.430926e-01
36     6.292690e-02
37    -6.255026e-03
38     2.104233e-01
39     4.752981e-02
40    -2.226199e-03
41     7.933319e-03
42     4.922129e-02
43     1.562703e-01
44     1.654176e-01
45    -1.805884e-01
46    -1.085233e-01
47     5.412136e-02
48    -4.954500e-03
49     1.242262e-01
50    -4.951696e-02
51    -1.094630e-01
52    -2.028550e-02
53    -6.561671e-02
54     1.649037e-01
55     6.545214e-02
56    -3.506217e-02
57    -1.203946e-02
58     2.223205e-01
59     9.231372e-02
60     1.744283e-01
61    -3.637285e-02
62     6.076528e-02
63     4.153151e-02
64     1.353915e-01
65    -1.069352e-01
66    -2.302818e-02
67    -3.677161e-02
68     7.366824e-02
69    -1.073434e-01
70    -1.601413e-01
71    -7.331805e-02
72     9.041051e-02
73     1.231124e-01
74    -1.114162e-01
75    -9.515034e-02
76     2.048974e-02
77     4.389499e-02
78     4.689618e-02
79    -8.072232e-02
80    -4.897787e-02
81     8.615039e-02
82     1.118210e-02
83     1.827232e-04
84     1.291876e-01
85     4.791930e-02
86    -1.561164e-01
87    -1.302685e-01
88    -9.005942e-02
89    -1.480637e-01
90     1.202116e-01
91    -4.768660e-02
92     4.448028e-02
93     1.583095e-01
94    -7.690951e-02
95     7.310532e-02
96    -3.996327e-02
97     1.641224e-01
98    -3.095109e-02
99     9.068620e-02
100    4.613448e-02
101   -3.453657e-03
102    8.810016e-02
103    3.886767e-02
104   -7.054850e-02
105   -5.651210e-02
106   -1.168088e-01
107    9.857541e-02
108   -4.724151e-02
109    5.771769e-02
110   -2.405190e-02
111   -1.432619e-02
112   -4.094599e-02
113   -2.257171e-02
114    1.138764e-01
115    9.988078e-03
116    5.702999e-02
117   -6.517232e-02
118   -3.786574e-02
119   -5.537427e-02
120    2.303855e-01
121   -8.175225e-02
122   -1.216491e-03
123    8.474145e-03
124    2.030437e-01
125   -2.179446e-01
126   -1.578859e-01
127   -1.661637e-01
128   -1.259029e-01
129    4.574648e-02
130    8.340736e-02
131   -1.557386e-03
132    9.818814e-02
133   -5.993004e-02
134    4.454617e-02
135   -7.363033e-02
136    2.311365e-01
137    3.177103e-02
138    1.023981e-01
139    2.564415e-02
140    1.663142e-02
141   -2.337696e-02
142    6.633113e-03
143   -1.048893e-01
144   -5.718889e-02
145    2.008070e-02
146   -8.238567e-02
147    3.337810e-02
148    1.077746e-02
149    2.016406e-02
150   -2.452729e-01
151    8.698673e-02
152    1.100125e-01
153    9.497721e-02
154   -1.424793e-02
155    3.324603e-02
156   -2.392609e-02
157   -3.543761e-02
158    3.212227e-02
159   -1.016631e-01
160    2.604744e-02
161    6.476328e-02
162    1.350631e-01
163    8.908737e-02
164   -4.544689e-02
165    4.988756e-02
166   -1.261699e-01
167    7.807012e-02
168    8.586696e-03
169    4.783699e-02
170    7.819753e-02
171   -2.511010e-01
172    5.354826e-02
173    1.525542e-01
174    2.701646e-02
175   -1.831799e-01
176   -2.889428e-02
177   -7.759524e-02
178    1.062651e-01
179   -7.763177e-03
180   -1.221566e-01
181   -7.496662e-02
182    6.194305e-02
183    5.873398e-02
184   -4.341222e-02
185   -5.803183e-02
186    2.393434e-01
187   -3.804233e-02
188    9.311978e-02
189    1.428793e-02
190    1.750944e-02
191   -2.398298e-03
192   -1.006524e-01
193    1.375191e-01
194   -1.681700e-02
195   -5.493797e-02
196    4.825121e-02
197   -8.191691e-03
198   -8.030937e-02
199    1.178579e-01
200   -3.986200e-02
201    1.835525e-01
202    2.846890e-01
203    2.067904e-03
204   -7.922766e-02
205    7.343257e-02
206   -6.324312e-02
207   -8.254665e-02
208    1.571749e-01
209    2.931703e-02
210   -3.922445e-02
211   -1.679612e-02
212    1.569857e-02
213    3.370002e-02
214    1.190539e-02
215    1.834415e-01
216   -1.804231e-01
217   -2.291029e-02
218   -4.368131e-02
219   -4.051499e-03
220    1.474241e-01
221   -7.028416e-02
222    9.177174e-02
223   -3.189593e-02
224   -1.110661e-01
225   -6.452027e-02
226    4.368454e-02
227    9.572649e-02
228    3.219917e-02
229    3.640267e-02
230    2.105604e-01
231   -2.076401e-01
232    4.430996e-02
233   -5.454177e-02
234   -2.784004e-03
235   -9.497608e-02
236   -2.147936e-02
237   -1.427147e-02
238    4.909257e-03
239    6.613189e-03
240   -3.280373e-03
241   -2.141271e-01
242    5.682020e-02
243   -8.674040e-02
244    1.360691e-01
245    3.835064e-02
246   -1.034684e-01
247    1.911102e-01
248    5.243248e-02
249    1.398704e-02
250    7.071142e-02
251    4.033527e-03
252   -3.626291e-02
253    3.091468e-02
254   -6.757922e-02
255   -6.124378e-03
256   -1.030869e-01
257   -3.490793e-02
258   -1.497294e-01
259   -2.437186e-01
260    1.088429e-01
261    8.192379e-02
262   -9.364399e-03
263   -1.995710e-02
264   -5.450911e-02
265   -1.337970e-01
266    8.275078e-02
267    5.981357e-02
268   -1.404618e-02
269   -4.301511e-02
270   -1.127719e-01
271    9.880089e-02
272    1.721709e-02
273   -1.016299e-01
274    4.729673e-02
275    2.056951e-02
276   -1.944836e-01
277    1.259625e-01
278    5.137198e-02
279    1.331034e-01
280   -7.363579e-02
281   -5.472502e-02
282   -3.776669e-02
283   -1.573532e-01
284   -2.020641e-01
285    1.201544e-01
286   -3.576184e-02
287   -3.774849e-02
288   -5.833574e-02
289   -1.951239e-02
290   -2.324062e-02
291   -2.052307e-01
292    8.823757e-02
293   -2.148019e-01
294    2.610066e-02
295   -6.622910e-02
296   -1.584768e-03
297    7.636259e-02
298   -5.300244e-03
299    2.574283e-02
300    5.624672e-02
301   -1.359460e-01
302    1.023995e-01
303   -9.445239e-02
304    1.192832e-01
305    9.937805e-02
306   -2.227314e-02
307   -6.185109e-03
308    1.045863e-01
309   -5.881690e-02
310    1.627165e-01
311    1.441759e-01
312   -2.102483e-03
313    9.995494e-03
314    8.071245e-03
315    1.981564e-02
316    5.362016e-02
317   -2.941310e-02
318   -8.606792e-02
319   -2.952266e-02
320    9.317468e-02
321   -1.977034e-01
322   -5.049206e-02
323   -2.272153e-02
324   -6.640623e-02
325   -7.463945e-02
326    6.238199e-02
327    4.061117e-02
328    4.240278e-03
329    8.983073e-02
330   -9.199369e-04
331    1.562215e-01
332   -8.764448e-03
333   -9.884495e-02
334    1.084070e-01
335    6.079465e-02
336   -4.689129e-02
337    8.761161e-02
338   -3.393935e-02
339    1.475869e-01
340    5.961895e-02
341   -9.271420e-02
342    5.523184e-02
343    3.526946e-02
344   -3.556719e-02
345    3.149887e-03
346    1.266702e-01
347    2.374862e-02
348   -6.697193e-02
349    4.110572e-02
350   -1.445165e-02
351   -5.106085e-02
352    4.918094e-02
353    9.212754e-02
354   -3.896695e-02
355    9.147592e-02
356   -1.479298e-01
357    1.658252e-01
358    9.362493e-02
359   -3.480597e-02
360   -3.301435e-02
361    7.646780e-02
362    4.676399e-02
363    2.092785e-02
364   -6.486538e-02
365   -7.987391e-02
366   -6.170743e-02
367   -1.666928e-01
368   -3.925381e-02
369   -1.074143e-02
370    3.295969e-02
371   -6.959830e-02
372    1.062541e-01
373   -2.324294e-02
374    4.458624e-02
375    7.494625e-02
376    3.023136e-02
377    8.337724e-02
378   -1.189307e-01
379   -1.150406e-01
380   -4.164489e-02
381   -1.195164e-01
382   -3.808668e-02
383   -4.555694e-02
384   -1.530553e-01
385   -1.784146e-01
386    2.936396e-02
387    2.737001e-02
388   -1.308759e-02
389    1.939796e-01
390    5.259317e-02
391    3.059345e-02
392   -7.614443e-02
393    4.657489e-02
394   -1.498004e-01
395   -1.923680e-02
396    9.855647e-02
397    5.563373e-02
398   -1.061731e-02
399    3.076606e-02
400    2.337907e-01
401    1.586159e-01
402    2.456125e-02
403   -5.941217e-02
404    3.050286e-03
405   -3.763476e-02
406    1.243391e-01
407    1.949630e-02
408    1.454117e-01
409    3.886513e-02
410   -6.494800e-02
411    8.021586e-02
412    8.823279e-02
413   -6.961517e-02
414   -4.065148e-02
415   -3.401314e-02
416   -5.467116e-02
417    9.492444e-02
418   -1.214461e-01
419    1.592518e-01
420   -4.585827e-02
421   -2.142396e-01
422    1.517716e-01
423   -7.857138e-02
424    2.740566e-02
425    6.469533e-03
426    3.049573e-02
427    1.738530e-01
428   -1.183981e-01
429   -9.926791e-02
430   -8.035544e-02
431   -4.480169e-03
432   -5.893043e-02
433   -1.566854e-01
434   -6.663567e-02
435    8.353755e-03
436   -1.276616e-01
437   -8.850563e-02
438    3.646297e-02
439    6.614164e-02
440   -1.386080e-01
441   -6.147051e-02
442   -3.151644e-02
443   -1.242393e-01
444    1.695919e-01
445    2.870591e-03
446   -5.839054e-02
447    2.419205e-02
448   -5.488889e-03
449   -2.486861e-02
450    9.818976e-02
451    2.248727e-01
452   -1.459026e-01
453    6.658970e-02
454    2.034688e-01
455    5.841585e-02
456    3.855831e-02
457    5.898424e-02
458    5.659366e-02
459   -3.933447e-03
460    7.367642e-03
461    5.677846e-02
462   -1.651747e-01
463    2.667993e-02
464    1.606905e-01
465   -2.024008e-01
466    2.662287e-01
467   -3.652815e-02
468    4.704766e-02
469    5.649099e-02
470   -7.966942e-02
471   -2.304953e-02
472    6.278219e-02
473   -6.239679e-02
474    1.035360e-01
475    4.490901e-02
476    7.129522e-02
477    1.544546e-02
478   -1.641609e-02
479   -1.146347e-01
480   -1.661256e-01
481    1.443602e-01
482    7.477594e-02
483    6.272408e-02
484   -1.162536e-02
485   -2.248989e-01
486   -2.176572e-01
487   -8.848702e-02
488   -1.412523e-02
489   -6.498494e-02
490   -6.415161e-02
491    1.108674e-01
492   -1.261409e-02
493   -5.465883e-02
494    3.345063e-02
495   -4.138326e-02
496    1.622364e-01
497   -8.805138e-02
498    9.574780e-02
499   -1.255496e-01
500   -1.600519e-01
501   -1.154444e-01
502   -3.474917e-02
503    6.752560e-02
504   -6.738858e-03
505    1.215880e-01
506    1.959904e-01
507    2.513453e-01
508    1.382920e-01
509    1.454943e-02
510    1.952504e-02
511    1.890510e-02
512    6.479796e-03
513   -3.760907e-02
514   -3.361211e-02
515    6.287680e-03
516   -1.997364e-01
517   -5.917415e-02
518    1.239165e-01
519   -9.765580e-02
520   -1.305188e-01
521   -2.922335e-02
522    3.008632e-02
523   -7.987198e-02
524    1.011500e-01
525    8.604286e-02
526   -2.385798e-01
527    2.410671e-02
528   -7.135752e-02
529   -1.899166e-02
530    3.212331e-02
531   -1.031421e-01
532    2.946364e-02
533   -5.147131e-02
534   -1.485357e-02
535   -6.552753e-02
536   -8.392589e-02
537    6.688623e-03
538   -3.209453e-03
539    6.715468e-02
540    2.333734e-02
541   -5.591362e-02
542   -1.144280e-01
543   -6.283689e-04
544   -8.933229e-02
545    3.060413e-01
546   -9.253083e-02
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1821  -3.137647e-03
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1823   8.689551e-02
1824   3.542071e-02
1825   1.148572e-01
1826   7.007449e-02
1827  -6.641797e-02
1828   2.525985e-02
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1833   2.048294e-02
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1836  -6.287377e-02
1837  -1.720711e-01
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1840   2.815763e-02
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1853   1.465185e-02
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1857   1.313620e-01
1858  -1.419360e-01
1859   1.596390e-01
1860   1.477453e-01
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1862  -1.392518e-02
1863   5.229069e-02
1864  -1.523721e-01
1865   3.586318e-02
1866  -2.122207e-03
1867  -5.037235e-02
1868   5.776385e-02
1869   4.733430e-02
1870  -1.001508e-01
1871   4.700605e-02
1872  -1.075130e-01
1873  -1.283614e-01
1874  -9.995595e-03
1875  -9.277816e-02
1876   3.210527e-02
1877   8.429355e-03
1878   7.786283e-02
1879   1.952278e-01
1880  -1.350314e-01
1881  -2.343535e-02
1882  -5.529597e-02
1883  -4.959765e-02
1884  -1.892479e-02
1885   1.785883e-02
1886   3.189399e-02
1887   2.994323e-02
1888   3.859111e-03
1889   7.398119e-02
1890   1.406754e-01
1891  -1.775209e-01
1892  -1.640800e-02
1893   1.965364e-02
1894   8.797136e-02
1895  -2.694035e-02
1896  -1.574023e-02
1897  -1.791486e-02
1898   2.922983e-02
1899  -6.736259e-02
1900   1.057861e-02
1901   7.544386e-02
1902  -4.303220e-02
1903  -5.060554e-02
1904  -2.388121e-02
1905  -6.506357e-04
1906   6.957807e-02
1907  -1.862376e-02
1908   4.138669e-02
1909  -5.417878e-02
1910   6.759408e-03
1911  -1.085755e-01
1912   7.446118e-02
1913  -5.211838e-03
1914   4.648572e-02
1915   1.082664e-01
1916   1.702668e-02
1917  -4.948076e-02
1918  -6.626676e-02
1919  -6.767814e-03
1920  -1.619350e-02
1921   8.026268e-02
1922  -4.444891e-02
1923  -9.837835e-02
1924  -4.402178e-02
1925  -3.330811e-02
1926  -1.427376e-01
1927  -3.410745e-02
1928   1.729015e-02
1929   9.918691e-02
1930   2.557800e-03
1931  -4.418522e-02
1932  -2.798803e-02
1933   1.174118e-02
1934   3.772234e-02
1935   9.479262e-02
1936  -8.439347e-02
1937   1.073567e-01
1938   1.479017e-01
1939   8.095173e-02
1940  -1.176574e-01
1941   6.915534e-02
1942  -8.194127e-02
1943  -2.484385e-03
1944  -7.371509e-02
1945   4.847556e-02
1946  -8.926689e-02
1947  -9.387125e-02
1948  -8.762187e-02
1949   5.366940e-02
1950   4.227986e-02
1951   1.895612e-01
1952   1.466803e-01
1953  -1.811383e-02
1954   2.319803e-02
1955   1.466327e-01
1956   9.051692e-02
1957  -5.329984e-02
1958   2.973054e-03
1959   4.909956e-02
1960   5.022870e-02
1961   8.800161e-02
1962  -2.187592e-01
1963   6.431454e-02
1964   4.640357e-02
1965  -8.927373e-02
1966   7.059601e-02
1967  -3.815025e-02
1968  -1.193992e-01
1969  -3.501274e-02
1970  -7.944152e-03
1971  -1.520476e-01
1972  -1.732135e-02
1973   9.011731e-02
1974  -2.003715e-01
1975  -1.745442e-01
1976   4.900905e-03
1977   2.460413e-02
1978   7.291301e-02
1979  -5.790940e-05
1980   6.594991e-02
1981   1.468910e-01
1982   5.925144e-02
1983  -9.389732e-02
1984  -1.878294e-01
1985   1.322180e-01
1986   2.421399e-02
1987   1.000518e-02
1988  -1.363164e-01
1989  -3.933809e-02
1990   1.934399e-01
1991  -4.696862e-02
1992   1.815589e-01
1993  -2.229144e-01
1994  -5.004471e-02
1995   1.300824e-02
1996   8.848830e-02
1997   1.864108e-02
1998  -1.717976e-02
1999  -8.841801e-02
2000  -9.774354e-02
2001   5.696497e-02
2002  -9.104224e-02
2003   1.371932e-02
2004   1.647088e-01
2005  -9.229532e-02
2006   2.315407e-01
2007  -4.499612e-02
2008  -1.332954e-02
2009  -8.319001e-02
2010   3.420235e-02
2011  -1.042377e-01
2012  -6.204188e-02
2013  -1.041652e-01
2014   3.373861e-02
2015  -6.573469e-03
2016   4.835347e-02
2017   1.293618e-01
2018   7.193430e-02
2019  -1.658518e-01
2020  -1.142284e-01
2021  -1.061329e-01
2022  -7.707957e-02
2023   3.618648e-02
2024  -1.212008e-01
2025  -8.346675e-02
2026   6.386581e-02
2027  -8.174775e-02
2028   1.394611e-02
2029  -4.704046e-02
2030  -5.972404e-02
2031  -1.718953e-02
2032   1.413224e-01
2033   7.313707e-03
2034  -1.438993e-01
2035  -5.006724e-02
2036  -3.492126e-02
2037  -3.247646e-02
2038   4.402239e-02
2039   1.462302e-02
2040  -8.198066e-02
2041   3.781987e-02
2042  -2.395809e-02
2043   1.781627e-01
2044   1.383603e-01
2045   5.376696e-02
2046   9.300524e-03
2047  -4.561968e-02
2048  -3.006662e-02
2049   4.811191e-02
2050  -6.415845e-02
2051   1.059835e-02
2052  -2.060844e-02
2053   7.775819e-02
2054   6.879269e-02
2055   9.996717e-02
2056  -1.309486e-01
2057  -2.675326e-02
2058  -1.124819e-01
2059  -2.276627e-01
2060   2.844593e-02
2061  -7.211789e-02
2062  -6.696773e-02
2063  -3.390791e-02
2064   9.906437e-02
2065  -4.733659e-02
2066   9.371415e-02
2067  -5.393975e-02
2068   2.895488e-02
2069   6.679436e-02
2070  -3.165398e-02
2071  -5.616047e-02
2072   4.635223e-02
2073  -5.827081e-02
2074   1.068718e-01
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2078  -2.536064e-02
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2080   1.170170e-01
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2082  -1.555448e-02
2083   1.646068e-01
2084   1.418306e-01
2085  -6.426246e-02
2086   1.069298e-01
2087  -8.026934e-02
2088  -5.078473e-02
2089  -1.007136e-01
2090  -2.127244e-01
2091  -1.302426e-01
2092   7.866740e-02
2093  -1.846444e-01
2094  -4.743612e-02
2095  -9.159239e-02
2096   3.582805e-01
2097   1.888245e-02
2098   1.257205e-01
2099  -9.028670e-02
2100  -1.171694e-01
2101  -1.889052e-02
2102   1.907217e-01
2103   1.293780e-02
2104   2.107245e-02
2105   4.843208e-02
2106   1.333883e-01
2107  -6.569721e-02
2108  -1.836590e-01
2109  -8.987027e-02
2110   3.652949e-02
2111   2.038628e-01
2112  -2.276614e-01
2113  -1.638915e-01
2114   1.462550e-01
2115   1.199068e-01
2116   6.548576e-02
2117   2.111201e-02
2118  -7.243407e-02
2119  -1.278674e-01
2120   3.694432e-02
2121  -1.438209e-01
2122  -8.429760e-02
2123   2.061392e-02
2124   1.495590e-01
2125  -7.184027e-03
2126  -7.660689e-02
2127   1.895560e-01
2128  -1.056319e-01
2129  -1.589939e-01
2130  -5.088882e-02
2131   2.982926e-02
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2134  -4.294258e-02
2135   5.074711e-02
2136   1.417370e-02
2137   8.537070e-02
2138   1.882645e-02
2139   4.817632e-02
2140   7.121769e-03
2141  -1.852808e-01
2142  -2.504726e-03
2143   9.494744e-02
2144  -1.106059e-02
2145   2.290448e-01
2146  -8.913885e-02
2147   5.733092e-02
2148  -1.399063e-02
2149   1.683963e-02
2150  -2.412058e-03
2151   1.465366e-01
2152   9.902131e-02
2153   1.199131e-01
2154   5.335442e-02
2155   2.276655e-02
2156   2.253415e-02
2157  -7.348643e-02
2158   1.147378e-01
2159  -1.989098e-01
2160   9.997475e-03
2161  -1.226040e-01
2162   3.091516e-03
2163   5.841228e-03
2164  -1.516145e-01
2165   3.832429e-02
2166   1.286521e-01
2167   5.407986e-02
2168  -3.046579e-02
2169   5.181677e-02
2170  -9.671268e-02
2171  -1.168579e-01
2172  -4.977893e-02
2173  -1.012012e-02
2174  -1.023838e-02
2175  -1.442088e-01
2176  -1.132333e-02
2177  -1.225659e-01
2178  -1.563950e-01
2179  -1.531322e-01
2180   3.684709e-02
2181  -1.453761e-01
2182   1.019134e-01
2183   5.428369e-02
2184  -1.489812e-01
2185   1.795082e-01
2186  -2.071930e-02
2187   1.463084e-03
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2200  -1.509091e-01
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2202   5.596972e-02
2203   1.794161e-01
2204   7.976258e-03
2205   1.093359e-01
2206   5.707174e-02
2207   1.051303e-01
2208   3.015275e-02
2209  -1.069251e-01
2210   9.155163e-02
2211   1.820812e-01
2212  -1.161170e-01
2213  -9.853762e-03
2214   8.767863e-02
2215   7.946260e-02
2216  -1.650806e-01
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2218   1.290526e-01
2219   5.943628e-02
2220  -8.239556e-02
2221  -1.225407e-01
2222   1.330732e-01
2223  -3.139811e-02
2224  -1.455899e-01
2225  -1.129529e-01
2226  -6.339191e-02
2227   3.120194e-02
2228  -5.665503e-02
2229   1.353036e-01
2230   1.926707e-01
2231  -8.638711e-02
2232   9.176368e-02
2233  -2.575433e-02
2234  -4.704774e-04
2235  -4.002000e-02
2236  -1.071033e-01
2237  -1.260678e-02
2238   2.885374e-02
2239   2.287822e-01
2240   6.570359e-02
2241   1.095367e-03
2242   3.227162e-02
2243  -8.887608e-03
2244  -2.731846e-04
2245   1.792869e-01
2246  -1.416961e-01
2247   7.489047e-02
2248  -1.761455e-02
2249   3.278019e-01
2250  -4.313114e-02
2251  -5.470507e-02
2252   9.870459e-02
2253  -7.285548e-02
2254  -1.069150e-01
2255   9.698764e-02
2256  -1.879181e-01
2257   3.231843e-02
2258  -9.493958e-02
2259  -1.467312e-01
2260   6.747602e-02
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2262   1.521107e-01
2263   2.877881e-02
2264   1.308621e-01
2265   5.662389e-02
2266   1.392784e-01
2267  -1.375053e-01
2268   9.699058e-02
2269   4.529990e-02
2270   5.373034e-03
2271   4.617688e-02
2272  -6.120562e-02
2273  -1.030595e-01
2274  -8.522982e-03
2275   1.087505e-01
2276   2.456238e-02
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2278   1.217213e-01
2279   8.020872e-02
2280  -1.206466e-01
2281  -7.976406e-03
2282   1.131194e-01
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2290   1.440131e-01
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2292   4.384519e-02
2293   2.681561e-02
2294   2.121593e-03
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2296   1.801678e-03
2297  -1.377260e-01
2298   1.622760e-01
2299   7.434816e-02
2300   1.361824e-01
2301   9.393603e-02
2302   2.781462e-02
2303   1.717383e-03
2304  -2.143151e-01
2305  -9.271265e-02
2306   8.553292e-02
2307  -4.191077e-02
2308   4.407320e-02
2309  -9.029608e-02
2310   7.285149e-02
2311  -8.547859e-02
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2313   5.305921e-03
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2319   1.006140e-01
2320  -6.239376e-02
2321  -1.011001e-01
2322  -8.880426e-02
2323  -2.981216e-02
2324  -1.719450e-02
2325  -7.839217e-02
2326   3.321677e-02
2327  -7.973697e-02
2328  -3.849962e-02
2329  -1.357218e-01
2330   2.735991e-01
2331   2.597519e-02
2332   4.436903e-02
2333  -6.674061e-02
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2335  -7.553624e-02
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2337   1.240426e-02
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99891  2.415776e-01
99892  4.115698e-02
99893  8.604102e-03
99894  6.709368e-02
99895  1.032013e-01
99896 -1.488352e-01
99897  1.674195e-02
99898 -1.015385e-01
99899  8.163711e-02
99900 -4.617193e-02
99901 -6.721857e-02
99902 -6.315001e-02
99903 -1.767369e-01
99904  1.724404e-01
99905  3.548980e-02
99906 -1.046146e-01
99907 -7.602382e-03
99908  1.415225e-01
99909  1.181187e-01
99910  4.519510e-02
99911 -5.651256e-02
99912 -6.407550e-02
99913  3.883809e-02
99914  3.236458e-02
99915 -6.669765e-02
99916 -1.017088e-01
99917  2.734089e-02
99918  1.889238e-01
99919 -6.251900e-03
99920 -7.917971e-02
99921  7.275828e-02
99922 -1.888739e-01
99923  4.312808e-03
99924 -9.340632e-02
99925 -1.082337e-01
99926 -3.652630e-02
99927 -1.262767e-01
99928 -5.249041e-02
99929 -8.680319e-02
99930  1.338522e-02
99931 -2.609433e-02
99932  9.651582e-02
99933  1.097025e-01
99934 -8.504531e-02
99935  1.322621e-01
99936 -6.072751e-02
99937 -1.240214e-01
99938  2.838048e-02
99939 -1.793973e-01
99940 -5.760987e-02
99941 -2.761886e-02
99942 -1.926246e-02
99943  1.419257e-01
99944  1.649803e-01
99945  9.795934e-02
99946  8.080340e-02
99947 -9.441622e-02
99948 -1.777448e-01
99949  9.760785e-02
99950 -1.112951e-01
99951  2.143850e-01
99952 -3.834805e-02
99953 -1.022695e-01
99954  4.215750e-02
99955  2.176569e-02
99956 -9.841115e-02
99957 -1.333775e-01
99958  1.089959e-01
99959  8.720990e-02
99960  1.052302e-01
99961 -5.703225e-02
99962  2.118778e-01
99963 -1.177021e-01
99964  1.268430e-02
99965  2.059512e-01
99966  4.181067e-02
99967  3.008193e-02
99968  1.731795e-01
99969  8.570697e-02
99970 -9.960302e-02
99971  8.752500e-02
99972  2.838680e-02
99973  1.147713e-01
99974 -7.375934e-02
99975 -1.754726e-01
99976 -4.172768e-02
99977  7.082608e-02
99978  1.874575e-02
99979 -9.618176e-02
99980 -1.618122e-02
99981 -8.221929e-02
99982  1.510601e-02
99983  8.368109e-03
99984  2.187968e-01
99985 -1.671268e-01
99986 -8.279772e-02
99987 -1.672344e-02
99988 -2.111730e-01
99989  1.288693e-02
99990 -1.001343e-01
99991 -9.341550e-02
99992  1.068084e-01
99993 -1.541918e-01
99994  6.003808e-03
99995 -4.627505e-02
99996  5.580505e-02
99997 -1.421165e-01
99998 -1.092917e-01
99999  2.281767e-01
 [ reached 'max' / getOption("max.print") -- omitted 1 rows ]

n <- 1000

medianaU <- replicate(1000, { 
  Xbarra <- median(runif(n,-1,1))
  Xbarra
})

df_simuls <- data.frame(medianaU = medianaU)

grafica6 <- ggplot(df_simuls) + 
  geom_histogram(aes(x = medianaU,
                     y = ..density..), bins=30L, size=1, fill = "#D39468", alpha = 0.5,) +

theme_minimal() 

n <- 10000

medianaU <- replicate(10000, { 
  Xbarra <- median(runif(n,-1,1)) 
  Xbarra
})

df_simuls <- data.frame(medianaU = medianaU)

grafica7 <- ggplot(df_simuls) + 
  geom_histogram(aes(x = medianaU,
                     
                     y = ..density..), bins=30L, size=1, fill = "#1BBD44", alpha = 0.5) +
  
  theme_minimal()
  
n <- 100000

medianaU <- replicate(100000, { 
  Xbarra <- median(runif(n,-1,1)) 
  Xbarra
})

df_simuls <- data.frame(medianaU = medianaU)

grafica8 <- ggplot(df_simuls) + 
  geom_histogram(aes(x = medianaU,
                     
                     y = ..density..), size=1, bins=30L, fill = "#BDB76B", alpha = 0.5) +
  theme_minimal()

2. Usando las funciones de la librería patchwork ponga en un mismo gráfico las cuatro gráficas de las funciones de densidad del inicio a)

(grafica10 | grafica6) / (grafica7 | grafica8)

3. ¿Qué densidad diría que tiene Xmed?

Se puede observar que con más simulaciones se parece a una distribución normal. Se puede decir que en la distribución normal: Mediana = Media

11. Considere X1, . . ., X999 variables aleatorias independientes e idénticamente distribuidas Unif [-1,1]. También considere X1000 - Unif[200,300] independiente de las otras 999 variables aleatorias, i.e. en total se tienen 1000 variables aleatorias independientes.

1. Para n = 100,1000,10000, 100000 haga n simulaciones para obtener la función de denisdad de probabilidad aproximada de X barra.

a <- -1
b <- 1
n <- 999
sim<- replicate(100000, {
  Xbarra <- sum(runif(n, min = a, max = b))
  Xbarra <- Xbarra + runif(1, min = 200, max = 300)
  Xbarra/1000
})

2. Usando las funciones de la librería patchworkponga en un mismo gráfico las cuatro fráficas de las funciones de masa del inicio a)

df <- data.frame(mediaa = sim[1:100])

grafica11 <- df%>% ggplot() + 
  geom_density(aes(x = mediaa), color = "#DA70D6", size = 1) + ggtitle("Densidad media aritmética n = 100") + ylab("f") + xlab("media aritmética") + 
  theme_minimal()


df <- data.frame(mediaa = sim[1:1000])

grafica1 <- df%>% ggplot() + 
  geom_density(aes(x = mediaa), color = "#B0C4DE", size = 1) + ggtitle("Densidad media aritmética n = 1000") + ylab("f") + xlab("media aritmética") + 
  theme_minimal()

df <- data.frame(mediaa = sim[1:10000])

grafica2 <- df%>% ggplot() + 
  geom_density(aes(x = mediaa), color = "#48D1CC", size = 1) + ggtitle("Densidad media aritmética n = 10000") + ylab("f") + xlab("media aritmética") + 
  theme_minimal()

df <- data.frame(mediaa = sim[1:100000])

grafica3 <- df%>% ggplot() + 
  geom_density(aes(x = mediaa), color = "#EEE8AA", size = 1) + ggtitle("Densidad media aritmética n = 100000") + ylab("f") + xlab("media aritmética") + 
  theme_minimal()

(grafica11 | grafica1) / (grafica2 | grafica3)

3. ¿Qué densidad diría que tiene XBarra?

Se puede observar que tiene forma de campana, entonces esta variable aleatoria tiene distribución normal con media en 0.25.

4. ¿Diría que se violenta el Teorema del Límite Central (TLC)?

Sí se violenta, ya que una de las condiciones del Teorema del Límite Central es que las variables aleatorias sean independientes e identicamente distribuidas y en este caso no se observa.

12. Considere X1, . . . , Xn variables aletorias independientes e idénticamente distribuidas, Xi-exp(1). Sea Xmed la mediana de estas n Xi´s, i.e.

                      Xmed = X(n+1/2) si n es impar
                            1/2(X(n/2) + X(n/2+1)) si n es par

1. Para n = 100,1000,10000,100000 haga 10,000 simulaciones para obtener la función de densidad de probabilidad aproximada de Xmed.

n <- 100

medianaE <- replicate(10000, { 
  Xbarra <- median(rexp(n,1))
  Xbarra
})

df_simuls <- data.frame(medianaE = medianaE)

grafica12 <- ggplot(df_simuls) +
  geom_histogram(aes( x= medianaE,
                      y = ..density..), bins=30L, size = 1, fill = "#FFD700", alpha = 0.5) +
  theme_minimal()

df_simuls

       medianaE
1     0.5268913
2     0.6933077
3     0.7935494
4     0.5608751
5     0.6640743
6     1.0044847
7     0.7240385
8     0.6436669
9     0.7897745
10    0.6663358
11    0.6432590
12    0.7720969
13    0.7243837
14    0.6573689
15    0.7942859
16    0.7003932
17    0.7726499
18    0.6902533
19    0.6686026
20    0.6245266
21    0.6396248
22    0.6254119
23    0.6194753
24    0.5130879
25    0.7554287
26    0.9075388
27    0.6724355
28    0.7225467
29    0.6486769
30    0.6785450
31    0.6193750
32    0.6215516
33    0.7486535
34    0.7489560
35    0.7416310
36    0.5809254
37    0.5654919
38    0.7343073
39    0.5408519
40    0.6472736
41    0.7590493
42    0.8497654
43    0.7558152
44    0.6219854
45    0.7703258
46    0.6073553
47    0.5932369
48    0.9763652
49    0.7646900
50    0.7479426
51    0.7613618
52    0.6321874
53    0.6972588
54    0.7171785
55    0.7332545
56    0.7335968
57    0.8826753
58    0.6816120
59    0.5957019
60    0.6783628
61    0.6155618
62    0.8212166
63    0.9208205
64    0.6383103
65    0.6459204
66    0.6044426
67    0.7717769
68    0.6957171
69    0.5137603
70    0.6090424
71    0.5703714
72    0.4659895
73    0.7515811
74    0.7550556
75    0.6838557
76    0.7611668
77    0.8203235
78    0.7150254
79    0.6240259
80    0.6515724
81    0.7786539
82    0.7456643
83    0.6267856
84    0.6540794
85    0.5409252
86    0.8156638
87    0.6575658
88    0.9692196
89    0.7899583
90    1.0440661
91    0.6811463
92    0.6534724
93    0.7150944
94    0.6220768
95    0.6926149
96    0.7510228
97    0.6491000
98    0.7450550
99    0.8193164
100   0.7393653
101   0.7249643
102   0.6436438
103   0.6868645
104   0.6039502
105   0.8253129
106   0.5415810
107   0.5729604
108   0.7799302
109   0.7747994
110   0.5380005
111   0.7814611
112   0.8829657
113   0.7350429
114   0.8050625
115   0.5634950
116   0.7761623
117   0.8536239
118   0.7589988
119   0.7498659
120   0.6343131
121   0.6701646
122   0.7475584
123   0.5911177
124   0.6050589
125   0.6710442
126   0.6680414
127   0.8121886
128   0.6808678
129   0.5817726
130   0.7120915
131   0.7800182
132   0.7011793
133   0.6940127
134   0.6857113
135   0.6410901
136   0.7351593
137   0.5461562
138   0.6579543
139   0.6534405
140   0.8626979
141   0.6131350
142   0.6781494
143   0.6559867
144   0.9314285
145   0.6764191
146   0.5052215
147   0.7599763
148   0.6943380
149   0.8537478
150   0.6663000
151   0.6094669
152   0.6412325
153   0.7540627
154   0.5692056
155   0.7946368
156   0.5653863
157   0.6523208
158   0.7732166
159   0.5586777
160   0.5045100
161   0.7134104
162   0.6080007
163   0.7568680
164   0.6678445
165   0.7054161
166   0.4870125
167   0.5846070
168   0.6591641
169   0.7685204
170   0.6507644
171   0.6893581
172   0.7795532
173   0.6920316
174   0.6827082
175   0.8079735
176   0.7192108
177   0.7030767
178   0.6389728
179   0.6997472
180   0.7708764
181   0.7807905
182   0.7320400
183   0.7742512
184   0.7513873
185   0.7044312
186   0.6551751
187   0.7271739
188   0.6783997
189   0.6804413
190   0.6825847
191   0.6260207
192   0.6056356
193   0.7736043
194   0.6074514
195   0.7067559
196   0.7457657
197   0.6304809
198   0.7057626
199   0.7337754
200   0.5832583
201   0.6958828
202   0.6123758
203   0.6404215
204   0.8030785
205   0.6308765
206   0.6882939
207   0.7058920
208   0.6842984
209   0.5740339
210   0.8227043
211   0.6249168
212   0.6945543
213   0.8297559
214   0.6020171
215   0.8855939
216   0.7623876
217   0.7947536
218   0.6461604
219   0.7903211
220   0.7934471
221   0.5987442
222   0.5304597
223   0.6297837
224   0.8405546
225   0.5894197
226   0.7895476
227   0.5867773
228   0.6352735
229   0.8952922
230   0.6447739
231   0.8342696
232   0.5655386
233   0.6768470
234   0.5100679
235   0.6464837
236   0.6748064
237   0.7836780
238   0.7482946
239   0.6279930
240   0.6715400
241   1.0258884
242   0.7325738
243   0.7065000
244   0.6401232
245   0.7912023
246   0.6470396
247   0.8846085
248   0.7228650
249   0.8013672
250   0.8317165
251   0.5719978
252   0.7730356
253   0.6482105
254   0.5456997
255   0.6450604
256   0.7079406
257   0.5708348
258   0.5115606
259   0.5389000
260   0.6837234
261   0.7518551
262   0.8551411
263   0.7115767
264   0.5799697
265   0.5238175
266   0.7385257
267   0.6399703
268   0.6411079
269   0.4990663
270   0.7536520
271   0.6851454
272   0.5994391
273   0.5930166
274   0.7137109
275   0.5938499
276   0.7332018
277   0.6976261
278   0.8328164
279   0.8282158
280   0.6094954
281   0.7853904
282   0.7294627
283   0.5465545
284   0.6625878
285   0.7169066
286   0.6593712
287   0.6408588
288   0.6006763
289   0.5092968
290   0.7559831
291   0.6603425
292   0.5692009
293   0.7825537
294   0.8505353
295   0.6274700
296   0.8778116
297   0.8317781
298   0.7387188
299   0.5162239
300   0.6366869
301   0.6257439
302   0.6920112
303   0.7281199
304   0.6139716
305   0.7680176
306   0.5869366
307   0.6790873
308   0.6615926
309   0.8022162
310   0.6768001
311   0.4183462
312   0.5479824
313   0.6650089
314   0.6887912
315   0.7287378
316   0.6644096
317   0.8348635
318   0.8079321
319   0.6784389
320   0.8341890
321   0.6242317
322   0.5871856
323   0.7250839
324   0.6145956
325   0.7657166
326   0.5927993
327   0.7026939
328   0.7188083
329   0.6965790
330   0.6312692
331   0.6738057
332   0.6496655
333   0.6169322
334   0.6851963
335   0.6638607
336   0.8413020
337   0.5914872
338   0.7465953
339   0.8390424
340   0.6286225
341   0.7091502
342   0.7828884
343   0.6213274
344   0.7843971
345   0.6652760
346   0.7642652
347   0.7524649
348   0.8376926
349   0.5435132
350   0.8052785
351   0.6085240
352   0.7661174
353   0.7416573
354   0.6694763
355   0.5819705
356   0.8139636
357   0.5165632
358   0.4929715
359   0.7580411
360   0.7178164
361   0.7123600
362   0.6836842
363   0.6134135
364   0.7295536
365   0.5866500
366   0.7697506
367   0.7785292
368   0.8675482
369   0.8505452
370   0.7288818
371   0.6301561
372   0.7131089
373   0.6383776
374   0.8531837
375   0.7399978
376   0.5838251
377   0.7380790
378   0.6611211
379   0.6558881
380   0.7190956
381   0.6843393
382   0.6153565
383   0.7630573
384   0.5041345
385   0.5767496
386   0.9346250
387   0.8096720
388   0.9092464
389   0.7004665
390   0.9212048
391   0.6166074
392   0.6550534
393   0.9660275
394   0.7351877
395   0.7173688
396   0.8259237
397   0.9610924
398   0.7767376
399   0.7896207
400   0.8165205
401   0.7361242
402   0.7554969
403   0.7103755
404   0.6671399
405   0.7762815
406   0.6921853
407   0.7381694
408   0.6879299
409   0.5723418
410   0.6279606
411   0.6699009
412   0.6600747
413   0.7707998
414   0.7408112
415   0.6959058
416   0.6896692
417   0.7480766
418   0.6220401
419   0.6389865
420   0.7355086
421   0.7289431
422   0.6621914
423   0.7115704
424   0.8282613
425   0.5655207
426   0.6289323
427   0.7140764
428   0.7209008
429   0.7323328
430   0.5632460
431   0.6082405
432   0.7289386
433   0.6810692
434   0.7653540
435   0.6304861
436   0.6722254
437   0.9083819
438   0.7706276
439   0.7889632
440   0.6879797
441   0.9400782
442   0.7894039
443   0.9100920
444   0.7227049
445   0.6669295
446   0.6305575
447   0.7784234
448   0.7586257
449   0.5995407
450   0.7198786
451   0.7789769
452   0.7004368
453   0.7002608
454   0.7291941
455   0.8107174
456   0.7871979
457   0.7154968
458   0.7497552
459   0.6403460
460   0.6547205
461   0.6096144
462   0.7382647
463   0.5609849
464   0.7424760
465   0.6419141
466   0.6467145
467   0.7257542
468   0.9807260
469   0.6556284
470   0.7894186
471   0.6511381
472   0.9336504
473   0.6948937
474   0.5736202
475   0.7244662
476   0.8275375
477   0.7006322
478   0.6792177
479   0.5589100
480   0.6470774
481   0.7633251
482   0.6375018
483   0.8275305
484   0.6612984
485   0.5049264
486   0.6128514
487   0.5582172
488   0.9867378
489   0.8804170
490   0.7947853
491   0.7720502
492   0.7225711
493   0.8045638
494   0.6785374
495   0.6465067
496   0.6566083
497   0.5178692
498   0.7697416
499   0.6920639
500   0.7523066
501   0.5851686
502   0.6554170
503   0.4950705
504   0.7624635
505   0.7634792
506   0.8024624
507   0.7660976
508   0.8221048
509   0.6337232
510   0.7408361
511   0.6837303
512   0.6625665
513   0.5895701
514   0.6609875
515   0.7357245
516   0.7987232
517   0.7552212
518   0.7676414
519   0.5757858
520   0.6482629
521   0.6294643
522   0.6211820
523   0.7331477
524   0.7655898
525   0.7081244
526   0.6997433
527   0.5323712
528   0.6757720
529   0.8200647
530   0.5465862
531   0.6829849
532   0.5768120
533   0.7334675
534   0.6163916
535   0.6868079
536   0.7604393
537   0.4483295
538   0.7812386
539   0.7533458
540   0.9094504
541   0.8099373
542   0.8049230
543   0.7196530
544   0.7745867
545   0.7131213
546   0.5831041
547   0.6372504
548   0.5581423
549   0.7019521
550   0.5298701
551   0.8324443
552   0.6147972
553   0.6666160
554   0.5538456
555   0.6298990
556   0.7122150
557   0.6927081
558   0.6382221
559   0.5631811
560   0.5608636
561   0.9064375
562   0.6147243
563   0.6430297
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9993  0.7685366
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9995  0.6385580
9996  0.6241574
9997  0.5935989
9998  0.6799187
9999  0.7179314
10000 0.8196255

n <- 1000

medianaE <- replicate(1000, { 
  Xbarra <- median(rexp(n,1))
  Xbarra
})

df_simuls <- data.frame(medianaE = medianaE)

grafica121 <- ggplot(df_simuls) + 
  geom_histogram(aes(x = medianaE,
                     y = ..density..), bins=30L, size=1, fill = "#FFB6C1", alpha = 0.5,) +

theme_minimal() 

n <- 10000

medianaE <- replicate(10000, { 
  Xbarra <- median(rexp(n,1)) 
  Xbarra
})

df_simuls <- data.frame(medianaE = medianaE)

grafica122 <- ggplot(df_simuls) + 
  geom_histogram(aes(x = medianaE,
                     
                     y = ..density..), bins=30L, size=1, fill = "#778899", alpha = 0.5) +
  
  theme_minimal()
  
n <- 100000

medianaU <- replicate(10000, { 
  Xbarra <- median(rexp(n,1)) 
  Xbarra
})

df_simuls <- data.frame(medianaE = medianaE)

grafica123 <- ggplot(df_simuls) + 
  geom_histogram(aes(x = medianaE,
                     
                     y = ..density..), size=1, bins=30L, fill = "#808800", alpha = 0.5) +
  theme_minimal()

2. Usando las funciones de la librería patchwork ponga en un mismo gráfico las cuatro gráficas de las funciones de densidad del inicio a)

(grafica12 | grafica121) / (grafica122 | grafica123)

3. ¿Qué densidad diría que tiene Xmed? Podría ser una exponencial, pero conforme se aumenta el nùmero de simulaciones, Xmed va tomando forma de una dsitribución normal, al igual que en el problema 10: Mediana = Media

13. Considere X1, . . ., Xn variables aleatorias indpendientes e idénticamente distribuidas, t(1).

1. Calcule teoricamente E(X)

e <- 3
n <- 100000
sim <- replicate(10000, {
  xbarra <- sum(rt(n, e))
  xbarra/n
})

2. Para n = 100,1000,10000,100000 haga 10,000 simulaciones para obtener la función de densidad de probabilidad aproximada de Xbarra, i.e. el promedio aritmético de las observaciones.

simulaciones <- data.frame(mediaa = sim[1:100])

grafica13 <- simulaciones %>% ggplot() + 
  
  geom_histogram(aes(x = mediaa, y = ..density..), color = "#B22222", size = 1, bins=30L) + ggtitle("Densidad media aritmética n = 100") + ylab("f") + xlab("media aritmética") +
  
  theme_minimal()

simulaciones <- data.frame(mediaa = sim[1:1000])

grafica131 <- simulaciones %>% ggplot() + 
  
  geom_histogram(aes(x = mediaa, y = ..density..), color = "#4B0082", size = 1, bins=30L) + ggtitle("Densidad media aritmética n = 1000") + ylab("f") + xlab("media aritmética") +
  
  theme_minimal()

simulaciones <- data.frame(mediaa = sim[1:10000])

grafica132 <- simulaciones %>% ggplot() + 
  
  geom_histogram(aes(x = mediaa, y = ..density..), color = "#FFB6C1", size = 1, bins=30L) + ggtitle("Densidad media aritmética n = 10000") + ylab("f") + xlab("media aritmética") +
  
  theme_minimal()

simulaciones <- data.frame(mediaa = sim[1:100000])

grafica133 <- simulaciones %>% ggplot() + 
  
  geom_histogram(aes(x = mediaa, y = ..density..), color = "#CD5C5C", size = 1, bins=30L) + ggtitle("Densidad media aritmética n = 100000") + ylab("f") + xlab("media aritmética") +
  
  theme_minimal()

3. Usando las funciones de la librería patchworkponga en un mismo gráfico las cuatro fráficas de las funciones de masa del inicio a)

(grafica13 | grafica131) / (grafica132 | grafica133)

Warning: Removed 90000 rows containing non-finite values (`stat_bin()`).

3. ¿Qué densidad diría que tiene Xbarra? Se puede observar que tiene forma de campana, entonces esta variable aleatoria tiene distribución normal con media 0.

4. ¿Diría que se violenta el Teorema del Límite Central (TLC)?

Sí se violenta, ya que una de las condiciones del Teorema del Límite Central es que las variables aleatorias sean independientes e identicamente distribuidas y en este caso no se observa.

14. Responda las siguientes preguntas:

1. Considere el lanzamiento de 2 dados y sea X la suma de sus valores. Lleve a cabo 100, 000 simulaciones y obtenga una aproximación de la función de masa de X.

sum_two_dice <-function() {
  roll <- sample(1:6, size = 2, replace = TRUE)
  sum(roll)
}

x <-replicate(100000, sum_two_dice())
df <- data.frame(suma = x)

grafica14 <- ggplot(df) +
  geom_histogram(aes(x = suma,
                     y =..count../sum(..count..)),
                 size = 1, bins = 30L, fill = "#20B2AA") + 
  ggtitle("F. Masa de la suma n = 100, 000") +
  ylab("f") + xlab("suma") + theme_light()

grafica14

2. Considere dos hexaedros, uno con todas las caras marcadas con “5”; el otro tiene 3 marcas de “2” y el resto de “6”. Simule el lanzamiento de estos dos dados y sea Y la suma de sus valores. Lleve a cabo 100,000 simulaciones y obtenga una aproximación de la función de masa de Y.

urn <- c(rep(2, 3), rep(6,1), rep(6,2))
sum_two_hex <- function() {
  roll <- sample(urn, size = 1, replace = TRUE)
  roll = roll + 5
}

x <- replicate(100000, sum_two_hex())
df <- data.frame(suma = x)

grafica141 <-ggplot(df) +
  geom_histogram(aes(x = suma,
                     y =..count../sum(..count..)),
                 size = 1, bins = 30L, fill = "#B0C4DE") +
  ggtitle("F. masa de la suma n = 100, 000") +
  ylab("f") + xlab("suma") +
  theme_light()

grafica141

3. Considere el lanzamiento de dos hexaedros, uno marcado con las etiquetas “1”, “2”, “2”, “3”, “3”, “4” en cada cara; el otro tiene las etiquetas “1”, “3”, “4”, “5”, “6” y “8”. Sea Z la suma de sus valores. Lleve a cabo 100, 000 simulaciones y obtenga una aproximación de la función masa de Z.

urn <- c(1, 2, 2, 3, 3, 4)
urn2 <- c(1, 3, 4, 5, 6, 8)
sum_two_hex <- function() {
  roll <- sample(urn, size = 1, replace = TRUE)
  roll2 <- sample(urn2, size = 1, replace = TRUE)
  sumaU = roll + roll2
}

x <- replicate(100000, sum_two_hex())
df <- data.frame(suma = x)

grafica142 <-ggplot(df) +
  geom_histogram(aes(x = suma,
                     y =..count../sum(..count..)),
                 size = 1, bins = 30L, fill = "#C71585") +
  ggtitle("F. masa de la suma n = 100, 000") +
  ylab("f") + xlab("suma") +
  theme_light()

grafica142

4. ¿Qué puede decir de las densidades/masas de X, Y y Z?

Las variables X y Z se parecen a una distribución Normal, gran cantidad de sus valores se concentran en la mediana.

Por otro lado, la variable Y tiene solo dos posibles resultados y están ubicados a lo extremos, la probabilidad es casi la misma para cada resultado.