Samuel Leidenberger Bitrán
Emiliano Bobadilla Franco
Asli Sandoval
library(ggplot2)
library(gridExtra)
library(reshape2)
library(patchwork)
library(dplyr)
Attaching package: 'dplyr'The following object is masked from 'package:gridExtra':
combineThe following objects are masked from 'package:stats':
filter, lagThe following objects are masked from 'package:base':
intersect, setdiff, setequal, union1. Considere \(X_1\), \(X_2\), . . . , \(X_n\) variables aleatorias independientes e idénticamente distribuidas
todas con distribución exponencial con media 5. También considere \(Y_1\), \(Y_2\), . . . , \(Y_m\) variables aleatorias independientes e idénticamente distribuidas todas con distribución exponencial con media 15 e independientes de los \(X_is\). Defina la cantidad
\[ T= \frac {1/n\sum_{i=1}^{n}X_i} {1/n\sum_{i=1}^{n}Y_i} \]
(a) Tome \(m = 20\). Simule \(n = 10, 100, 1000, 1000\) \(X_is\) y \(Y_js\) y sugiera una posible distri-
bución.
(b) Tome \(n = 20\). Simule \(m = 10, 100, 1000, 1000\) \(X_is\) y \(Y_js\) y sugiera una posible distri-
bución.
simular_T <- function(n, m) {
X <- rexp(n, rate = 1/5)
Y <- rexp(m, rate = 1/15)
T <- (sum(X) / n) / (sum(Y) / m)
return(T)
}
n_values <- c(10, 100, 1000, 10000)
m_values <- c(10, 100, 1000, 10000)
num_simulations <- 1000Creamos una lista para almacenar los gráficos
graficos <- list()
for (i in 1:length(n_values)) {
n <- n_values[i]
# Gráficos para m = 20
T_n <- replicate(num_simulations, simular_T(n, m = 20))
df_n <- data.frame(n = paste0("n=", n), T = T_n)
graficos[[paste0("n_", n, "_m_20")]] <- ggplot(df_n, aes(x = T, fill = n)) +
geom_histogram(alpha = 0.6, bins = 50, position = 'identity') +
theme_minimal() +
labs(title = paste0("n = ", n, ", m = 20"), x = "T", y = "Frecuencia")
# Gráficos para n = 20
T_m <- replicate(num_simulations, simular_T(n = 20, m = n_values[i]))
df_m <- data.frame(m = paste0("m=", n_values[i]), T = T_m)
graficos[[paste0("n_20_m_", n_values[i])]] <- ggplot(df_m, aes(x = T, fill = m)) +
geom_histogram(alpha = 0.6, bins = 50, position = 'identity') +
theme_minimal() +
labs(title = paste0("n = 20, m = ", n_values[i]), x = "T", y = "Frecuencia")
}Mostramos los gráficos en una cuadrícula 4x2
grid.arrange(grobs = graficos, ncol = 2)
(c) Encuentre la distribución de \(T\)
Observamos que nuestra \(T\) tiene un comportamiento similar a una distribución Cauchy
2. Considere \(X_1\), \(X_2\), . . . , \(X_n\) variables aleatorias independientes e idénticamente distribuidas todas con distribución normal estándar. Defina
\[ U= \frac {\sqrt{n}[X_1+X_2+...+X_n] }{X_1^2+X_2^2+...+X_n^2} \]
Haga \(n = 10, 100, 1000, 10000\) simulaciones de \(U\) y sugiera una posible distribución para \(U\)
simulate_U <- function(n, num_simulations) {
U_values <- numeric(num_simulations)
for (i in 1:num_simulations) {
X <- rnorm(n)
U <- sqrt(n) * sum(X) / sqrt(sum(X^2))
U_values[i] <- U
}
return(U_values)
}
num_simulations_list <- c(10, 100, 1000, 10000)
n_values <- c(10, 100, 1000)
all_data <- data.frame()
for (n in n_values) {
for (num_simulations in num_simulations_list) {
U_values <- simulate_U(n, num_simulations)
data <- data.frame(n = n, num_simulations = num_simulations, U = U_values)
all_data <- rbind(all_data, data)
}
}
all_data_melted <- melt(all_data, id.vars = c("n", "num_simulations"), variable.name = "Variable", value.name = "U")
ggplot(all_data_melted, aes(x = U, fill = factor(num_simulations))) +
geom_histogram(alpha = 0.5, position = "identity", bins = 30) +
facet_wrap(~ n) +
labs(title = "Histograma de U para diferentes n y número de simulaciones",
x = "Valor de U",
y = "Frecuencia",
fill = "Num. de Simulaciones") +
theme_minimal()
Observamos que tiene apariencia de normal estándar
3. Considere \(X_1, X_2, . . . , X_n\) variables aleatorias independientes e id ́enticamente distribuidas
todas con distribución \(Unif (0, 5)\). Defina
\[ U = máx[X_1,X_2,...,X_n]-5 \]
Haga \(n = 10, 100, 1000, 10000\) simulaciones de \(U\) y sugiera una posible distribución para \(U\)
Definimos la función de simulación de \(U\)
simular_u <- function(n, num_simulaciones = 1000) {
u <- rep(0, num_simulaciones)
for (i in 1:num_simulaciones) {
xi <- runif(n, min = 0, max = 5)
u[i] <- max(xi) - 5
}
return(u)
}Valores de \(n\) para simular
valores_n <- c(10, 100, 1000, 10000)Simulamos \(U\) y almacenamos en un marco de datos
u_datos <- data.frame()
for (n in valores_n) {
u_simulado <- simular_u(n)
u_n <- data.frame(n = rep(n, length(u_simulado)), u = u_simulado)
u_datos <- rbind(u_datos, u_n)
}Usamos melt para transformar los datos para crear el gráfico de facetas
u_melted <- melt(u_datos, id.vars = "n")Creamos gráfico de facetas
ggplot(u_melted, aes(x = value)) +
geom_histogram(bins = 50, fill = "blue", color = "black") +
facet_wrap(~ n, ncol = 2, scales = "free_x") +
ggtitle("Histogramas de U para diferentes valores de n") +
xlab("Valor de U") + ylab("Frecuencia")
Observamos que arece una distribución exponencial recorrida e invertida
4. Se lanzan 3 dados balanceados. Sea \(X\) la suma de las caras
(a) Para \(n = 100, 1000, 10000, 100000\) haga \(n\) simulaciones para obtener la función de
masa de probabilidad aproximada de \(X\)
Función para realizar \(n\) simulaciones y calcular la función de masa de probabilidad aproximada de \(X\)
simulaciones <- function(n) {
resultados <- replicate(n, sum(sample(1:6, size = 3, replace = TRUE)))
freq_absoluta <- table(resultados)
freq_relativa <- freq_absoluta / n
return(freq_relativa)
}
(b) Usando las funciones de las librería patchwork y ggplot2 ponga en un mismo gr ́afico
las cuatro gráficas de las funciones de masa del inicio anterior.
Creamos las gráficas para \(n = 100, 1000, 10000, 100000\)
n_valores <- c(100, 1000, 10000, 100000)
graficas <- lapply(n_valores, function(n) {
datos <- as.data.frame(simulaciones(n))
colnames(datos) <- c("X", "Probabilidad")
p <- ggplot(datos, aes(x = as.numeric(as.character(X)), y = Probabilidad)) +
geom_bar(stat = "identity", fill = "blue", alpha = 0.6) +
ggtitle(paste("n =", n)) +
labs(x = "X", y = "Probabilidad") +
theme_minimal()
return(p)
})Combinamos las gráficas en un solo frame
grafica_combinada <- graficas[[1]] / graficas[[2]] | graficas[[3]] / graficas[[4]]
print(grafica_combinada)
(c) Para \(n = 10, . . . , 100000\) haga \(n\) simulaciones para obtener un estimado de la proba-
bilidad \(P(X ≤ 3)\). Ponga en un gráfico \(n\) en el eje horizontal y la probabilidad en el
eje vertical.
Estimamos la probabilidad \(P(X <= 3)\) para \(n = 10, …, 100000\)
n_simulaciones <- c(100,1000,10000,100000)
probabilidad <- sapply(n_simulaciones, function(n) {
resultados <- simulaciones(n)
p_x_menor_igual_3 <- sum(resultados[names(resultados) %in% c("3")])
return(p_x_menor_igual_3)
})Graficamos \(n\) en el eje horizontal y la probabilidad en el eje vertical
datos_probabilidad <- data.frame(n = n_simulaciones, probabilidad = probabilidad)
p_probabilidad <- ggplot(datos_probabilidad, aes(x = n, y = probabilidad)) +
geom_line(color = "red") +
labs(x = "n", y = "Probabilidad P(X ≤ 3)") +
theme_minimal()
print(p_probabilidad)
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 n ́umeros en las bolas.
(a) Para \(n = 100, 1000, 10000, 100000\) haga \(n\) simulaciones para obtener la función de
masa de probabilidad aproximada de \(X\).
urna_simulacion <- function(n) {
urna <- 1:7
resultados <- numeric(n)
for (i in 1:n) {
muestra <- sample(urna, 2, replace = FALSE)
suma <- sum(muestra)
resultados[i] <- suma
}
return(resultados)
}Realizamos simulaciones para \(n = 100, 1000, 10000, 100000\)
n100 <- urna_simulacion(100)
n1000 <- urna_simulacion(1000)
n10000 <- urna_simulacion(10000)
n100000 <- urna_simulacion(100000)
(b) 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.
crear_histograma <- function(datos, titulo) {
df <- data.frame(x = datos)
ggplot(df, aes(x)) +
geom_histogram(binwidth = 1, color = "black", fill = "lightblue", boundary = 0.5) +
scale_x_continuous(breaks = seq(min(datos), max(datos), 1)) +
labs(title = titulo, x = "Suma de las bolas (X)", y = "Frecuencia") +
theme_minimal()
}Creamos los histogramas
hist_n100 <- crear_histograma(n100, "Histograma de n = 100")
hist_n1000 <- crear_histograma(n1000, "Histograma de n = 1000")
hist_n10000 <- crear_histograma(n10000, "Histograma de n = 10000")
hist_n100000 <- crear_histograma(n100000, "Histograma de n = 100000")Organizamos los histogramas en una matriz 2x2 utilizando patchwork
(hist_n100 | hist_n1000) / (hist_n10000 | hist_n100000)
(c) Para \(n = 10, . . . , 100000\) haga \(n\) simulaciones para obtener un estimado de la proba-
bilidad \(P(X ≤ 10)\). Ponga en un gráfico \(n\) en el eje horizontal y la probabilidad en el
eje vertical.
calcular_probabilidad <- function(datos) {
exitos <- sum(datos <= 10)
probabilidad <- exitos / length(datos)
return(probabilidad)
}Realizamos las simulaciones y calculamos las probabilidades
valores_n <- c(10, 100, 1000, 10000, 100000)
probabilidades <- numeric(length(valores_n))
for (i in 1:length(valores_n)) {
resultados <- urna_simulacion(valores_n[i])
probabilidades[i] <- calcular_probabilidad(resultados)
}Creamos un gráfico de probabilidades
df <- data.frame(n = valores_n, probabilidad = probabilidades)
ggplot(df, aes(x = n, y = probabilidad)) +
geom_point() +
geom_line() +
scale_x_log10() +
labs(title = "Probabilidad P(X ≤ 10) en función de n",
x = "n (cantidad de simulaciones)", y = "Probabilidad P(X ≤ 10)") +
theme_minimal()
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 números en las bolas.
Definimos la función de simulación
simulacion <- function(n) {
sumas <- numeric(n)
for (i in 1:n) {
bola1 <- sample(1:7, 1, replace = TRUE)
bola2 <- sample(1:7, 1, replace = TRUE)
suma <- bola1 + bola2
sumas[i] <- suma
}
return(sumas)
}
(a) Para \(n = 100, 1000, 10000, 100000\) haga \(n\) simulaciones para obtener la función de
masa de probabilidad aproximada de \(X\).
Realizamos las simulaciones y obtenemos la función de masa de probabilidad aproximada
n_values <- c(100, 1000, 10000, 100000)
resultados <- lapply(n_values, simulacion)
fmp_aprox <- lapply(resultados, function(x) table(x) / length(x))
(b) 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.
Creamos las gráficas de las funciones de masa
graficas <- lapply(1:4, function(i) {
ggplot(data.frame(x = as.numeric(names(fmp_aprox[[i]])), y = as.vector(fmp_aprox[[i]])), aes(x = x, y = y)) +
geom_col(fill = "blue") +
labs(title = paste("n =", n_values[i]), x = "Suma", y = "Probabilidad") +
theme_minimal()
})Poner las gráficas en un mismo frame
grafico_combinado <- graficas[[1]] / graficas[[2]] / graficas[[3]] / graficas[[4]]
print(grafico_combinado)
(c) Para \(n = 10, . . . , 100000\) haga \(n\) simulaciones para obtener un estimado de la proba-
bilidad \(P(X ≤ 10)\). Ponga en un gráfico \(n\) en el eje horizontal y la probabilidad en el
eje vertical.
Estimamos la probabilidad \(P(X <= 10)\) para distintos valores de \(n\)
n_seq <- c(100, 1000, 10000, 100000)
prob_estimada <- sapply(n_seq, function(n) {
sumas <- simulacion(n)
mean(sumas <= 10)
})Creamos el gráfico de \(n\) en el eje horizontal y la probabilidad en el eje vertical
ggplot(data.frame(n = n_seq, prob = prob_estimada), aes(x = n, y = prob)) +
geom_line() +
labs(title = "Estimación de P(X <= 10)", x = "n", y = "Probabilidad") +
theme_minimal()
7. En un salón de clases del curso de Cálculo de Probabilidad 2 hay 50 estudiantes. Cada
estudiante pone en papel su clave única (CU) y lo introduce en una urna común. Poste-
riormente dichxs estudiantes toman un papel de la urna. Sea \(X\) el número de alumnos que
sacaron el papelito con su clave única.
(a) Para \(n = 100, 1000, 10000, 100000\) haga \(n\) simulaciones para obtener la función de
masa de probabilidad aproximada de \(X\).
simulaciones <- function(n) {
resultados <- numeric(n)
for (i in 1:n) {
estudiantes <- 1:50
urna <- sample(estudiantes, size = 50, replace = FALSE)
coincidencias <- sum(estudiantes == urna)
resultados[i] <- coincidencias
}
return(resultados)
}
n_vals <- c(100, 1000, 10000, 100000)
prob_mass_func <- list()
for (n in n_vals) {
sim <- simulaciones(n)
prob_mass_func[[as.character(n)]] <- table(sim) / n
}(b) 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.
plot_prob_mass_func <- function(prob_mass_func, n) {
df <- data.frame(NumEstudiantes = as.numeric(names(prob_mass_func)),
Probabilidad = as.numeric(prob_mass_func))
ggplot(df, aes(x = NumEstudiantes, y = Probabilidad)) +
geom_col() +
labs(title = paste0("n = ", n),
x = "Número de estudiantes que sacaron su propia clave",
y = "Probabilidad") +
theme_minimal()
}plot_100 <- plot_prob_mass_func(prob_mass_func[['100']], 100)
plot_1000 <- plot_prob_mass_func(prob_mass_func[['1000']], 1000)
plot_10000 <- plot_prob_mass_func(prob_mass_func[['10000']], 10000)
plot_100000 <- plot_prob_mass_func(prob_mass_func[['100000']], 100000)(plot_100 | plot_1000) / (plot_10000 | plot_100000)
(c) Para \(n = 10, . . . , 100000\) haga \(n\) simulaciones para obtener un estimado de la proba-
bilidad \(P(X ≤ 15)\). Ponga en un gráfico \(n\) en el eje horizontal y la probabilidad en el
eje vertical.
prob_x_menorigual_15 <- function(n) {
sim <- simulaciones(n)
sum(sim <= 15) / n
}
n_vals_c <- c(10,1000,10000,100000)
prob_vals_c <- sapply(n_vals_c, prob_x_menorigual_15)
df_c <- data.frame(n = n_vals_c, probabilidad = prob_vals_c)
ggplot(df_c, aes(x = n, y = probabilidad)) +
geom_line() +
labs(title = "Probabilidad P(X ≤ 15) en función de n",
x = "Número de simulaciones (n)",
y = "Probabilidad P(X ≤ 15)") +
theme_minimal()
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 sumando requeridos para alcanzar dicho objetivo. Por ejemplo
si obtuvo los números \(0.35, 0.58, 0.22\) se tiene que \(N = 3\) (pues se necesitó 3 sumandos
para que la suma sea mayor ó igual que 1).
(a) Para \(n = 100, 1000, 10000, 100000\) haga \(n\) simulaciones para obtener la funci ́on de
masa de probabilidad aproximada de \(N\).
(b) Usando las funciones de las librería patchwork y ggplot2 ponga en un mismo gr ́afico
las cuatro gráficas de las funciones de masa del inicio anterior.
(c) Para \(n = 10, . . . , 100000\) 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
Función para simular \(N\) sumandos
simular_sumandos <- function() {
suma <- 0
n <- 0
while (suma < 1) {
suma <- suma + runif(1)
n <- n + 1
}
return(n)
}Función para calcular la función de masa de probabilidad aproximada de \(N\)
calcular_fmp <- function(n) {
simulaciones <- replicate(n, simular_sumandos())
fmp <- table(simulaciones) / n
fmp_df <- as.data.frame(fmp)
fmp_df$simulaciones <- as.numeric(as.character(fmp_df$simulaciones))
fmp_df$Freq <- as.numeric(fmp_df$Freq)
return(fmp_df)
}Función para graficar la función de masa de probabilidad
graficar_fmp <- function(data, n) {
ggplot(data, aes(x = simulaciones, y = Freq)) +
geom_col(fill = "dodgerblue", width = 0.7) +
labs(title = paste0("FMP para n = ", n),
x = "Sumandos",
y = "Frecuencia") +
theme_minimal()
}Simulamos para diferentes valores de \(n\)
valores_n <- c(100, 1000, 10000, 100000)
fmp_data <- lapply(valores_n, calcular_fmp)
fmp_plots <- lapply(seq_along(fmp_data), function(i) graficar_fmp(fmp_data[[i]], valores_n[i]))Graficamos las FMP en un solo gráfico
combined_plot <- fmp_plots[[1]] / fmp_plots[[2]] / fmp_plots[[3]] / fmp_plots[[4]]
print(combined_plot)
Estimamos el valor esperado de \(E(N)\) para diferentes valores de \(n\) y los graficamos
estimar_valor_esperado <- function(data) {
sum(data$simulaciones * data$Freq)
}
valores_esperados <- sapply(fmp_data, estimar_valor_esperado)
ggplot(data.frame(n = valores_n, E_N = valores_esperados), aes(x = n, y = E_N)) +
geom_point() +
geom_line() +
scale_x_log10() +
labs(title = "Valores esperados de E(N) en función de n",
x = "n",
y = "E(N)") +
theme_minimal()
9. Considere \(X1, . . . , Xn\) variables aleatorias independientes e idénticamente distribuidas,
\(X_i∼Unif(0, 1)\). Sea \(X(2)\) el segundo valor mas de pequeño de estas \(n X_i’s\)
Función para realizar las simulaciones
simular_x2 <- function(n, n_simulaciones) {
x2_vals <- numeric(n_simulaciones)
for (i in 1:n_simulaciones) {
xi_vals <- runif(n)
x2_vals[i] <- sort(xi_vals)[2]
}
return(x2_vals)
}Realizamos las simulaciones y guardamos los resultados
n_valores <- c(100, 1000, 10000, 100000)
n_simulaciones <- 10000
simulaciones <- lapply(n_valores, function(n) simular_x2(n, n_simulaciones))Creamos un data.frame con los resultados
datos <- data.frame()
for (i in 1:length(n_valores)) {
df <- data.frame(x2 = simulaciones[[i]], n = n_valores[i])
datos <- rbind(datos, df)
}Creamos y combinamos las gráficas
p1 <- ggplot(datos[datos$n == 100,], aes(x = x2)) + geom_density() + labs(title = "n = 100")
p2 <- ggplot(datos[datos$n == 1000,], aes(x = x2)) + geom_density() + labs(title = "n = 1000")
p3 <- ggplot(datos[datos$n == 10000,], aes(x = x2)) + geom_density() + labs(title = "n = 10000")
p4 <- ggplot(datos[datos$n == 100000,], aes(x = x2)) + geom_density() + labs(title = "n = 100000")
(p1 | p2) / (p3 | p4)
(a) Para \(n = 100, 1000, 10000, 100000\) haga 10, 000 simulaciones para obtener la función
de densidad de probabilidad aproximada de \(X(2)\).
# Establece una semilla para obtener resultados reproducibles (opcional)
set.seed(123)
eo100=c()
eo1000=c()
eo10000=c()
eo100000=c()
# Genera n valores aleatorios uniformes en el intervalo (0,1)
for(i in 0:10000){
n100 <- runif(100)
n1000 <- runif(1000)
n10000 <- runif(10000)
n100000 <- runif(100000)
# Ordena los valores generados
ordenados100 <- sort(n100)
ordenados1000 <- sort(n1000)
ordenados10000 <- sort(n10000)
ordenados100000 <- sort(n100000)
eo100<-append(eo100,ordenados100[2])
eo1000<-append(eo1000,ordenados1000[2])
eo10000<-append(eo10000,ordenados10000[2])
eo100000<-append(eo100000,ordenados100000[2])
}
(b) Usando las funciones de la librería patchwork ponga en un mismo gráfico las cuatro
gráficas de las funciones de masa del inicio (a).
crear_histograma <- function(datos, titulo) {
df <- data.frame(x = datos)
ggplot(df, aes(x)) +
geom_histogram(binwidth = .001, color = "black", fill = "lightblue", boundary = 0.5) +
scale_x_continuous(breaks = seq(min(datos), max(datos), 1)) +
labs(title = titulo, x = "(X)", y = "Frecuencia") +
theme_minimal()
}
length(eo100000)[1] 10001# Creamos histogramas
hist_n100 <- crear_histograma(eo100, "Histograma de n = 100")
hist_n1000 <- crear_histograma(eo1000, "Histograma de n = 1000")
hist_n10000 <- crear_histograma(eo10000, "Histograma de n = 10000")
hist_n100000 <- crear_histograma(eo100000, "Histograma de n = 100000")
# Organizamos los histogramas en una matriz 2x2 utilizando patchwork
(hist_n100 | hist_n1000) / (hist_n10000 | hist_n100000)
(c) ¿Qué densidad diría que tiene \(X(2)\)?
Parece una beta
(d) ¿Es consistente este resultado con ejemplo teórico que se vió en clase?
Si, una \(beta(2,n-2+1)\)
10. Considere \(X1, . . . , Xn\) variables aleatorias independientes e idénticamente distribuidas,
\(Xi ∼ Unif (−1, 1)\). Sea \(Xmed\) la mediana de estas \(n\) \(X_i’s\), i.e.
\[ X_{med}= \begin{cases} X_{\frac{n+1}{2}} & \text{si n es impar}\\ \frac{1}{2}[X_\frac{n}{2}+X_{\frac{n}{2}+1} & \text{si n es par } \end{cases} \]
mediana <- function(x) {
n <- length(x)
if (n %% 2 == 0) {
return((x[n/2] + x[n/2 + 1])/2)
} else {
return(x[(n+1)/2])
}
}Definimos la función de densidad de \(X_{med}\)
densidadXmed <- function(n, N) {
resultados <- numeric(N)
for (i in 1:N) {
muestra <- runif(n, -1, 1)
med <- mediana(muestra)
resultados[i] <- med
}
density(resultados)
}(a) Para \(n = 100, 1000, 10000, 100000\) haga 10, 000 simulaciones para obtener la función
de densidad de probabilidad aproximada de \(Xmed\).
Obtener la función de densidad de \(X_{med}\) para \(n = 100, 1000, 10000, 100000\)
n <- c(100, 1000, 10000, 100000)
N <- 10000
densidades <- lapply(n, densidadXmed, N=N)
(b) Usando las funciones de la librería patchwork ponga en un mismo gráfico las cuatro
gráficas de las funciones de masa del inicio (a).
Graficamos las funciones de densidad en un mismo gráfico
p <- NULL
for (i in 1:length(n)) {
p[[i]] <- ggplot(data.frame(x=densidades[[i]]$x, y=densidades[[i]]$y),
aes(x=x, y=y)) + geom_line() +
labs(title=paste0("n=", n[i]))
}
wrap_plots(p)
(c) ¿Qué densidad diría que tiene \(Xmed\)?
La densidad de \(X_{med}\) se asemeja a una distribución normal
(a) Para \(n = 100, 1000, 10000, 100000\) haga \(n\) simulaciones para obtener la función de
densidad de probabilidad aproximada de \(\bar{X}\).
(b) Usando las funciones de la librería patchwork ponga en un mismo gráfico las cuatro
gráficas de las funciones de masa del inicio (a).
10000,100000 es computacionalmente intensivo
dist_unif <-c()
for(i in 1:1000) {
dist_unif <- append(dist_unif,runif(999, min = -1, max = 1))
dist_unif <-append(dist_unif,runif(1, min = 200, max = 300))
}
medias100=c()
for(i in 1:100) {
medias100<-append(medias100,sum(dist_unif[i*1000-999:i*1000])/1000)
}
medias1000=c()
for(i in 1:1000) {
medias1000<-append(medias1000,sum(dist_unif[i*1000-999:i*1000])/1000)
}crear_histograma <- function(datos, titulo) {
df <- data.frame(x = datos)
ggplot(df, aes(x)) +
geom_histogram(binwidth = 10, color = "black", fill = "lightblue", boundary = 0.5) +
scale_x_continuous(breaks = seq(min(datos), max(datos), 1)) +
labs(title = titulo, x = "(X)", y = "Frecuencia") +
theme_minimal()
}hist_n100 <- crear_histograma(medias100, "Histograma de n = 100")
hist_n1000 <- crear_histograma(medias1000, "Histograma de n = 1000")
# Crearmos histogramas
medias1000 [1] 0.7569436 1.0492000 1.3218237 1.5492232 1.7784019 2.0755630
[7] 2.3256072 2.5842839 2.8666379 3.1455624 3.4446875 3.7157679
[13] 4.0042071 4.2218851 4.4963361 4.7196930 4.9831399 5.2742565
[19] 5.5295949 5.7478205 6.0070543 6.2535432 6.4567126 6.7494330
[25] 7.0320029 7.2601767 7.5537836 7.8365467 8.0741688 8.3276314
[31] 8.5407811 8.7449699 9.0058233 9.2339008 9.5259838 9.7295481
[37] 10.0151341 10.2330270 10.5269299 10.7324254 10.9945173 11.2728538
[43] 11.5437040 11.8293050 12.1226119 12.3467019 12.6454632 12.9264610
[49] 13.2086500 13.5054617 13.7571190 13.9868313 14.1903914 14.4234580
[55] 14.6950619 14.9085721 15.1244065 15.3815894 15.5899657 15.8802925
[61] 16.1245601 16.3897285 16.6633320 16.8819049 17.1105950 17.4064554
[67] 17.6260406 17.8770569 18.1762160 18.4570485 18.7330227 18.9528160
[73] 19.2205365 19.4746471 19.6821463 19.9084073 20.1551452 20.3594927
[79] 20.5621590 20.8301217 21.0393853 21.2914349 21.5844262 21.8518879
[85] 22.1151692 22.3762049 22.6358782 22.8380071 23.0899922 23.3572965
[91] 23.6474701 23.8496949 24.0766560 24.2954952 24.5206811 24.8082564
[97] 25.0345749 25.2617855 25.5206385 25.7820386 26.0144951 26.2236135
[103] 26.4415517 26.6488310 26.9404221 27.1501822 27.4423652 27.6498350
[109] 27.8864745 28.1338935 28.3848337 28.6698158 28.8868525 29.0926262
[115] 29.3595220 29.5648396 29.8046360 30.0900052 30.3261673 30.5410400
[121] 30.8208900 31.0797829 31.3533602 31.5832119 31.8021576 32.0810445
[127] 32.3532205 32.6014692 32.8811336 33.1671829 33.3975689 33.6608308
[133] 33.9141740 34.1799839 34.4026110 34.6554895 34.8558686 35.1348611
[139] 35.3652564 35.5756026 35.7931442 36.0406577 36.2604161 36.5470024
[145] 36.7500028 37.0168169 37.2422842 37.4570850 37.7139959 37.9582275
[151] 38.1946446 38.4782744 38.6970435 38.9108543 39.1116641 39.4026466
[157] 39.7011476 40.0009535 40.2416080 40.4492437 40.6535883 40.9064465
[163] 41.1427209 41.4094058 41.6749623 41.9196285 42.1392941 42.3789552
[169] 42.6569081 42.9169637 43.1358880 43.4149232 43.7043499 43.9850894
[175] 44.2461232 44.5388162 44.7691532 45.0282345 45.3223455 45.5749545
[181] 45.8532864 46.0769414 46.3128138 46.5949996 46.8032288 47.0687027
[187] 47.3544148 47.5668142 47.8582757 48.1274565 48.3403532 48.6317620
[193] 48.8509966 49.0826729 49.3019450 49.5131878 49.7773532 50.0482519
[199] 50.3083663 50.5151451 50.7602987 51.0373565 51.3292508 51.6065814
[205] 51.8745165 52.0903193 52.3313606 52.6140924 52.8884952 53.1130554
[211] 53.4024984 53.6887154 53.9465579 54.2355510 54.5266588 54.7312192
[217] 54.9784534 55.2592894 55.4885547 55.7609980 55.9655545 56.2112408
[223] 56.4814410 56.7366447 56.9777102 57.2717964 57.5025249 57.7920624
[229] 58.0504880 58.2550111 58.4762651 58.6807702 58.9489377 59.2066620
[235] 59.5046900 59.7805483 60.0302810 60.3163682 60.5216064 60.8183749
[241] 61.0626895 61.3464092 61.6463947 61.9432608 62.1493210 62.3628282
[247] 62.6462808 62.8581353 63.0731326 63.3023217 63.5573750 63.7713330
[253] 64.0259085 64.2417916 64.5321507 64.7535758 64.9807918 65.1822632
[259] 65.3930220 65.6673964 65.8920857 66.1062515 66.3965461 66.6157177
[265] 66.8968786 67.1743954 67.4515956 67.6949261 67.9197043 68.1959693
[271] 68.4313229 68.6403291 68.9118336 69.1891122 69.4677021 69.6725141
[277] 69.8837042 70.1350653 70.3795026 70.6084317 70.8361354 71.0426288
[283] 71.3348030 71.5900661 71.8165139 72.0952907 72.3793436 72.6612946
[289] 72.8826148 73.1113028 73.3585308 73.6384592 73.8977813 74.1930085
[295] 74.4880664 74.6949852 74.9109394 75.1324402 75.3957720 75.6790068
[301] 75.9454022 76.2104458 76.4128282 76.6165214 76.8612531 77.1334157
[307] 77.3778640 77.6522694 77.9321923 78.2190826 78.4501515 78.6717816
[313] 78.8931615 79.1658902 79.4177966 79.6786128 79.8913021 80.1124823
[319] 80.3762822 80.5818987 80.8193711 81.0448602 81.3316315 81.5577993
[325] 81.7975406 82.0196888 82.2623388 82.5076615 82.7525120 82.9658215
[331] 83.2466827 83.5382205 83.8179241 84.0807735 84.2871961 84.5580830
[337] 84.8013332 85.0908761 85.3387222 85.5963547 85.8443837 86.0808085
[343] 86.3713305 86.5921383 86.8302410 87.0988319 87.3151145 87.5714398
[349] 87.8228992 88.0388901 88.2518588 88.4648195 88.7522784 89.0157696
[355] 89.2606698 89.5489108 89.8008147 90.0916022 90.3251356 90.6112564
[361] 90.8858689 91.1708635 91.4542275 91.6786510 91.9460125 92.2015925
[367] 92.4259223 92.6510944 92.9429496 93.1990713 93.4559660 93.7427766
[373] 94.0374748 94.3214279 94.5571768 94.8392094 95.0496310 95.2788898
[379] 95.5211648 95.7351329 95.9844621 96.2750294 96.5311091 96.8277947
[385] 97.0314957 97.2518084 97.4931680 97.7593741 98.0115582 98.2539598
[391] 98.5401789 98.8321428 99.0991199 99.3110197 99.5903980 99.8540231
[397] 100.1302899 100.3830775 100.6673863 100.9142922 101.1395190 101.4058415
[403] 101.6957088 101.9097713 102.1883234 102.4752872 102.7259414 102.9830116
[409] 103.2143564 103.4506634 103.6723785 103.9570862 104.2492813 104.5163987
[415] 104.7796965 104.9959125 105.2540817 105.4695788 105.6791643 105.9570091
[421] 106.1600820 106.4041401 106.6124475 106.8549827 107.1240740 107.4066912
[427] 107.6920459 107.9598552 108.2551560 108.5023508 108.7137367 108.9314037
[433] 109.2134076 109.4513083 109.7499961 109.9564176 110.2494019 110.4815905
[439] 110.7345622 111.0133773 111.2187063 111.5033622 111.7378178 112.0003345
[445] 112.2302594 112.5266664 112.7670342 112.9908990 113.2265114 113.4520356
[451] 113.7138411 113.9764341 114.2288713 114.5031416 114.7806336 114.9954330
[457] 115.2924217 115.5036809 115.8036154 116.0650670 116.2839818 116.5819708
[463] 116.8702763 117.0820568 117.3218932 117.5630581 117.7775692 118.0524502
[469] 118.2614528 118.5330462 118.7765464 119.0155843 119.2629182 119.5082909
[475] 119.7444607 119.9917356 120.2433859 120.5029421 120.7441107 121.0117000
[481] 121.2802432 121.5174497 121.7709290 122.0472060 122.2631732 122.5362546
[487] 122.7409085 122.9508708 123.2040268 123.4812549 123.7720729 123.9949985
[493] 124.2178450 124.4944920 124.7912729 125.0550759 125.2658054 125.5031515
[499] 125.7458025 126.0354481 126.2617288 126.5031817 126.7404512 126.9847360
[505] 127.2722540 127.5331480 127.8076419 128.0816742 128.3750917 128.6738843
[511] 128.9444211 129.1700208 129.3975843 129.6658826 129.8819338 130.1445255
[517] 130.3796151 130.6140377 130.8596132 131.1024176 131.3492281 131.5546467
[523] 131.7978749 132.0949567 132.3867109 132.6749555 132.9327866 133.1398783
[529] 133.3864071 133.6580366 133.9288974 134.1422557 134.4061659 134.6936461
[535] 134.9447892 135.2063524 135.5002433 135.7875116 136.0019753 136.2620890
[541] 136.5335692 136.7623120 136.9653224 137.1917592 137.4125275 137.7003081
[547] 137.9449820 138.1869465 138.4541878 138.7146462 138.9615587 139.1672206
[553] 139.3989340 139.6380376 139.9240377 140.2227442 140.5214668 140.7712044
[559] 141.0101407 141.2779723 141.5759089 141.8096647 142.0581071 142.2808340
[565] 142.5684773 142.8344175 143.0900157 143.3752652 143.6297143 143.9087650
[571] 144.1207046 144.3290968 144.5569560 144.8053447 145.0813864 145.3702004
[577] 145.6564421 145.8857146 146.1856347 146.4688166 146.7617222 147.0143743
[583] 147.3032600 147.5887314 147.8530914 148.1290178 148.4053118 148.7042910
[589] 148.9829982 149.1900643 149.4618943 149.7188026 149.9216770 150.1861409
[595] 150.4840138 150.7547249 150.9794465 151.2025795 151.4921774 151.7202613
[601] 151.9582978 152.2023897 152.4090208 152.6749147 152.9066736 153.1736392
[607] 153.4704403 153.7675190 154.0426510 154.2898409 154.5384363 154.7815329
[613] 155.0460662 155.3248082 155.5409486 155.7487066 156.0117838 156.2848886
[619] 156.5209778 156.7881505 157.0546445 157.2880739 157.4990283 157.7540424
[625] 158.0285422 158.2820088 158.5488469 158.8149660 159.1034393 159.3458640
[631] 159.5672293 159.7875508 160.0873529 160.3277534 160.5834944 160.8414174
[637] 161.1246771 161.4132758 161.6946866 161.9248014 162.1849422 162.4680784
[643] 162.7046149 162.9850516 163.2486440 163.4647279 163.6827137 163.8874796
[649] 164.0894154 164.3320583 164.5607885 164.7884289 165.0021795 165.2636209
[655] 165.5616494 165.7939888 166.0836380 166.3129088 166.6109425 166.8232370
[661] 167.0678840 167.2878118 167.5856858 167.8686811 168.1676722 168.4086456
[667] 168.6246765 168.8534236 169.0685287 169.3015982 169.5625851 169.8006380
[673] 170.0164146 170.2780597 170.5324091 170.8077131 171.0252299 171.3212700
[679] 171.6151930 171.8719592 172.1086902 172.3693999 172.6051603 172.8072955
[685] 173.1068609 173.3077550 173.5259195 173.8116637 174.0613453 174.3190432
[691] 174.5699195 174.8374614 175.1183379 175.4091139 175.6323908 175.8727421
[697] 176.1426522 176.4157129 176.6672669 176.9185766 177.1187804 177.3666884
[703] 177.6018633 177.8256235 178.0375641 178.2908018 178.5402438 178.7689183
[709] 178.9838942 179.2602982 179.5179747 179.7953195 180.0287638 180.2413976
[715] 180.4834830 180.7259909 180.9565546 181.1628870 181.4074783 181.6822105
[721] 181.9475648 182.2170124 182.4772210 182.7160837 182.9635955 183.2270578
[727] 183.5248326 183.7485075 183.9950640 184.2932678 184.5681178 184.8400910
[733] 185.1319414 185.4232306 185.6520551 185.8909616 186.1730370 186.3850841
[739] 186.6693094 186.8973573 187.1936787 187.4068261 187.6387548 187.8544830
[745] 188.1417049 188.4246330 188.7209070 188.9267169 189.1943814 189.4930755
[751] 189.7117590 190.0061869 190.2652057 190.5417415 190.7603219 190.9856596
[757] 191.2677309 191.4703224 191.7626844 192.0219911 192.3053943 192.5273509
[763] 192.7361470 192.9788695 193.2492472 193.5458821 193.7684156 194.0628510
[769] 194.3613931 194.6019867 194.8241792 195.1053482 195.3580111 195.5837815
[775] 195.8611873 196.0860732 196.3560324 196.5751478 196.8514204 197.0958995
[781] 197.3283992 197.5943956 197.8903985 198.1133127 198.3411177 198.6342781
[787] 198.9040635 199.1421106 199.3613984 199.6456818 199.9320696 200.1676704
[793] 200.3740056 200.6126377 200.9102726 201.1388916 201.4178197 201.6190631
[799] 201.8309236 202.0395073 202.2692659 202.5664839 202.7761820 203.0183404
[805] 203.2950991 203.5646502 203.8242047 204.0487289 204.2964804 204.5078543
[811] 204.7712240 205.0176917 205.3176002 205.5713589 205.8109980 206.0490341
[817] 206.2730323 206.5511481 206.8043414 207.1039687 207.3587034 207.5685409
[823] 207.7883677 208.0412131 208.2964483 208.5501522 208.7877124 209.0178284
[829] 209.2219678 209.4479634 209.7314385 209.9406464 210.1776105 210.4563112
[835] 210.6867067 210.9363642 211.1963879 211.4244150 211.6397713 211.8902689
[841] 212.1193507 212.4027880 212.6938916 212.9829444 213.2246593 213.4601555
[847] 213.7437858 214.0384863 214.3335450 214.5478963 214.7732195 215.0636710
[853] 215.2952245 215.5455025 215.8049509 216.0464518 216.3348233 216.5913753
[859] 216.8718769 217.1111042 217.3534506 217.5704090 217.8145040 218.0311448
[865] 218.2805263 218.5452307 218.8090695 219.0195370 219.2731458 219.5626114
[871] 219.8434674 220.0852371 220.3410573 220.6018549 220.8754123 221.1224605
[877] 221.3582198 221.6447062 221.8576377 222.1241468 222.3397833 222.6106979
[883] 222.8845206 223.1804933 223.4059669 223.6146861 223.8167638 224.0537869
[889] 224.3224915 224.5582767 224.8282635 225.1048784 225.3561406 225.6306410
[895] 225.9122164 226.1859797 226.4454803 226.7062086 227.0043031 227.2621871
[901] 227.4655372 227.7171551 227.9520339 228.2111397 228.4810527 228.7788495
[907] 229.0612113 229.2903201 229.5487264 229.8030562 230.0034707 230.2662374
[913] 230.5207618 230.7761555 230.9962923 231.1999041 231.4879791 231.7352027
[919] 231.9402244 232.2053419 232.5045786 232.7866286 233.0659972 233.3538458
[925] 233.5555330 233.8303042 234.1175477 234.3948596 234.6297611 234.8971489
[931] 235.1514675 235.4502356 235.6566098 235.9226423 236.1281934 236.3857572
[937] 236.6247948 236.8267871 237.1128364 237.3802004 237.6621862 237.9256950
[943] 238.1272256 238.4103058 238.6627775 238.8678567 239.0859677 239.3209303
[949] 239.5673088 239.8407872 240.0436417 240.2553261 240.4990250 240.7312874
[955] 240.9973733 241.2758313 241.5736504 241.7908364 242.0672506 242.3028611
[961] 242.5847444 242.8555567 243.1180958 243.3977313 243.6647960 243.9215074
[967] 244.1679517 244.4443850 244.6452848 244.8592444 245.0829369 245.3317544
[973] 245.5743815 245.7857438 246.0604807 246.2941934 246.5854290 246.8532008
[979] 247.0704179 247.3040969 247.5290738 247.7412884 247.9876068 248.2170772
[985] 248.4492984 248.6657455 248.9620328 249.2552099 249.4965387 249.7775653
[991] 250.0452462 250.2510218 250.4784208 250.7228329 250.9874852 251.2618108
[997] 251.5364929 251.8013666 0.0000000 0.2380837# Organizamos los histogramas en una matriz 2x2 utilizando patchwork
(hist_n100 | hist_n1000) 
(c) ¿Qué densidad diría que tiene \(\bar{X}\)?
Parece una densidad uniforme
(d) ¿Diría que se violenta el Teorema del Límite Central?
No, ya que no son distribuciones idénticas
12. Considere \(X_1, . . . , X_n\) variables aleatorias independientes e idénticamente distribuidas,
\(X_i ∼ exp(1)\). Sea \(Xmed\) la mediana de estas \(n\) \(X_i’s\), i.e. \[
X_{med}= \begin{cases} X_{\frac{n+1}{2}} & \text{si n es impar}\\ \frac{1}{2}[X_\frac{n}{2}+X_{\frac{n}{2}+1} & \text{si n es par } \end{cases}
\]
Definimos la función de mediana
mediana <- function(x) {
n <- length(x)
if (n %% 2 == 0) {
return((x[n/2] + x[n/2 + 1])/2)
} else {
return(x[(n+1)/2])
}
}Definir la función de densidad de \(X_{med}\)
densidadXmed <- function(n, N) {
resultados <- numeric(N)
for (i in 1:N) {
muestra <- rexp(n, 1)
med <- mediana(muestra)
resultados[i] <- med
}
density(resultados)
}(a) Para \(n = 100, 1000, 10000, 100000\) haga 10, 000 simulaciones para obtener la función
de densidad de probabilidad aproximada de \(Xmed\).
Obtener la función de densidad de \(X_{med}\) para \(n = 100, 1000, 10000, 100000\)
n <- c(100, 1000, 10000, 100000)
N <- 10000
densidades <- lapply(n, densidadXmed, N=N)(b) Usando las funciones de la librería patchwork ponga en un mismo gráfico las cuatro
gráficas de las funciones de masa del inicio (a).
p <- NULL
for (i in 1:length(n)) {
p[[i]] <- ggplot(data.frame(x=densidades[[i]]$x, y=densidades[[i]]$y),
aes(x=x, y=y)) + geom_line() +
labs(title=paste0("n=", n[i]))
}
wrap_plots(p)
(c) ¿Qué densidad diría que tiene \(Xmed\)?
La densidad de \(X_{med}\) se asemeja a una distribución normal con media igual a \(log(2)\) y varianza igual a \(\pi^{2/6}\)
13. Considere \(X_1, . . . , X_n\) variables aleatorias independientes e idénticamente distribuidas, \(t(1)\).
(a) Calcule teoricamente \(E(X)\)
Teóricamente \(E(X)=0=med(X)=moda(X)\)
(b) Para \(n = 100, 1000, 10000, 100000\) haga 10, 000 simulaciones para obtener la función
de densidad de probabilidad aproximada de \(\bar{X}\), i.e. el promedio aritmético de las
observaciones.
# Establecer la semilla para la reproducibilidad de los resultados
set.seed(123)
med_arit<-c()
for(n in c(100,1000,10000,100000)){
for(i in 1:1000){
# Número de valores a simular
df <- 1 # Grados de libertad
# Simular valores de la distribución t de Student
t_values <- rt(n, df)
med_arit<-append(med_arit,sum(t_values)/n)
}
}
med_arit [1] 4.133642e+00 1.214052e+00 -6.473249e-01 1.153164e+00 1.362228e+00
[6] 1.003327e-01 5.228007e-01 1.055026e+02 -4.409836e+00 1.410580e+00
[11] -7.721249e-01 1.334293e+00 -4.620658e-01 1.311404e+01 -1.149182e+00
[16] 1.427828e+00 2.575894e-01 -1.940338e-01 -5.583881e+00 1.425002e+00
[21] -8.316089e+00 -1.444350e-01 -3.327594e+00 -3.992560e-01 1.999578e+00
[26] 6.166858e+00 -3.739182e+00 -4.972474e+00 6.143137e-01 1.634351e+00
[31] 1.753571e+00 2.787898e+00 4.681753e-01 -1.646720e+00 3.355847e+00
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[1791] 2.536309e+00 6.656854e-01 1.180997e+00 -8.751125e-01 3.863820e-01
[1796] -4.851594e+00 -8.726801e-02 3.209573e-01 -3.034656e+00 8.849612e+00
[1801] 6.747373e+00 -2.681391e-01 -2.788802e+01 2.360071e-01 -3.211052e+00
[1806] -9.313890e-02 4.327340e+00 -1.417870e-01 9.138666e-01 1.110055e+00
[1811] -1.586061e+00 -1.189378e+00 -2.713589e+00 3.495186e-01 7.306870e-01
[1816] 8.124745e-01 -2.156832e+00 2.065987e-01 3.056295e+00 3.885726e+00
[1821] -7.845206e-01 -5.374409e+00 -1.133318e+00 -2.822940e+00 -5.929194e-01
[1826] -5.983842e-01 1.476116e+00 -2.925957e+00 -3.640208e-02 -1.374352e+00
[1831] -2.265816e-01 -4.810125e-01 -5.687178e-01 -9.113804e-01 3.178856e+01
[1836] -1.194178e+00 2.391717e+00 -1.057243e+00 -1.594302e+00 6.569956e-02
[1841] -1.262243e+00 5.284985e-01 -9.333968e-01 1.244959e+00 -1.454971e+00
[1846] -4.934559e-01 -1.527758e-01 -1.789261e-01 -4.101315e-01 -8.463818e-01
[1851] 3.073086e-01 -2.387619e+00 -2.033327e-01 9.347308e-03 -7.596913e-01
[1856] 2.362249e-01 -7.536272e-01 3.765633e+01 2.138059e+01 1.283422e+00
[1861] -1.908418e+00 9.997264e+00 -3.085821e-01 2.409238e+00 -7.670772e+00
[1866] 7.552045e+00 -1.256300e-01 -2.050395e+00 4.608787e-01 4.289894e-01
[1871] 6.646852e-01 5.724803e-01 3.912277e+00 -8.072783e-01 5.994430e-01
[1876] -1.011607e+01 -3.028354e-01 8.309129e-01 -7.639892e-01 9.152423e-01
[1881] 2.398053e+00 6.871904e-02 6.407940e-01 1.601909e+00 -8.950256e-01
[1886] 1.585273e+00 -1.298006e+00 -1.389231e+00 9.366890e+00 -3.389125e-01
[1891] 6.317154e-01 -1.323465e+00 -8.135333e-01 1.948408e-01 3.025829e-01
[1896] 6.155512e-01 1.966021e-01 -7.629072e-01 1.713525e+00 -7.225831e-01
[1901] -4.054766e+00 8.114823e-02 -1.503201e+01 3.609502e+00 -6.082425e+00
[1906] 1.780841e-01 2.444702e+00 -9.266974e-01 9.664476e-01 1.218929e+00
[1911] 3.903774e-02 8.715959e-01 -1.445504e-01 -1.596701e+00 8.155158e-02
[1916] 7.379162e-01 7.195825e-01 -1.070582e+00 -1.783908e+00 7.089787e-01
[1921] -1.043488e+00 -4.093101e-01 8.390263e-01 -1.927460e+00 -3.764600e-01
[1926] 1.570875e+00 -1.989254e+00 -5.943785e-01 -8.899083e-01 -6.733044e-02
[1931] -3.615410e+00 1.493245e+00 6.362756e+00 2.392458e-01 -1.238134e+01
[1936] 1.353417e+00 -3.844173e+00 -1.952006e+01 3.361529e-01 -2.491194e+00
[1941] 1.245370e-01 -6.134820e-01 -2.745981e+00 6.501214e-01 6.373867e-03
[1946] 1.601660e+00 4.666071e+00 2.955438e+00 9.246004e-02 7.718320e+00
[1951] 1.071499e+00 -1.611064e-01 3.085253e+00 6.185615e-01 -1.478259e+00
[1956] 2.337929e+00 1.377982e+00 -1.250216e+00 4.573685e-01 7.587108e-01
[1961] -2.776100e-01 -5.175874e+01 -8.641220e-01 -9.670877e-01 -5.706198e-01
[1966] -6.097956e-01 -4.679053e+01 2.807598e-01 -3.218104e-01 -7.790571e-02
[1971] 2.769717e-01 -4.174086e-01 3.040710e+00 -1.708693e+00 2.744364e+00
[1976] 2.357203e-01 4.365521e-01 -9.221430e-01 8.070762e+00 2.099746e+00
[1981] -8.402787e-02 5.977318e-01 1.784846e+00 -2.085202e-01 -8.733864e-01
[1986] -8.462970e-01 -5.903058e+00 -2.284765e-01 -7.702429e-02 4.758529e-01
[1991] 1.588117e+01 8.028088e-01 2.021347e-02 6.689348e+00 -9.767739e-01
[1996] -2.186503e+01 1.115676e+00 2.679515e-01 3.252536e-01 -8.553696e-02
[2001] 1.111674e+00 5.235531e-01 -3.434568e-01 -5.013793e+00 3.818247e+00
[2006] -6.513532e+00 1.325581e+00 -5.339886e+00 -2.878471e+00 -5.084372e+00
[2011] 3.826583e-01 -1.716207e+00 1.256916e+00 5.926651e+00 1.211415e+00
[2016] -1.022434e+00 -9.063731e+00 -1.079491e+00 3.309042e+00 -1.070196e+00
[2021] -3.842265e+00 -1.570838e+00 3.217643e+00 -1.826609e+00 3.587322e-01
[2026] 9.672458e-01 -1.834945e+00 1.567952e+00 5.528170e+00 4.715087e+00
[2031] 4.762741e+00 -2.157865e+00 2.201263e-02 1.353619e+00 1.230807e-01
[2036] 1.554314e+00 5.916899e-01 8.014567e-01 2.045352e-01 1.949876e+01
[2041] 1.485471e+00 -4.651289e+00 1.373981e+01 -1.431523e-02 1.189836e+00
[2046] -5.227658e-01 4.931729e-01 8.425788e-01 -8.259255e+00 -1.545345e+00
[2051] 5.310998e-01 -1.651657e+00 4.101933e-01 -1.839587e+00 1.392193e-01
[2056] -1.144568e+01 -2.540639e+00 -1.455204e+00 1.288773e-01 2.507838e+00
[2061] -3.713551e-01 -2.162938e-01 -2.145924e+00 -3.509798e-01 1.242361e+01
[2066] 5.250171e-01 -1.455512e+01 5.803383e-01 1.931623e-01 -4.335726e+00
[2071] 4.694927e-01 2.359008e+00 2.556874e+01 1.932899e-01 5.791214e-01
[2076] -1.621189e+00 3.952416e+00 1.296664e+00 2.185121e-01 6.994976e-01
[2081] 1.153983e+00 -3.324471e-01 -2.460752e-02 -2.886875e+00 -5.031491e-01
[2086] 1.691147e+00 1.133717e+00 -1.668991e+00 6.198206e+00 -2.986730e+01
[2091] -1.478215e+00 1.856355e+00 -7.767276e-01 -1.490836e+00 4.622314e-01
[2096] -5.052491e-01 5.420291e-01 -1.836686e+00 9.632047e-01 9.625885e+00
[2101] -3.328017e-01 -1.548174e+00 -7.228379e-01 -1.489684e-02 -3.111759e+00
[2106] -2.805802e+00 1.114501e+00 -3.884595e-01 -1.465454e+00 -7.543600e-01
[2111] 1.533163e+00 2.583283e-01 -1.133535e-03 1.642396e+00 2.369276e-02
[2116] 6.069423e-01 2.041837e-01 5.688097e-01 3.186968e+00 -1.605943e+00
[2121] -1.437319e+00 -1.158757e+00 1.565784e-01 4.908447e-01 9.569406e-01
[2126] -1.479647e+00 -1.427666e+00 -4.617388e+00 1.227220e+00 -2.309635e-01
[2131] -2.281222e+00 -1.688470e+01 1.269062e+00 -1.012167e+00 1.281981e+00
[2136] -2.560217e+00 6.947185e+00 1.838154e-01 9.155317e-01 5.899559e-01
[2141] -1.143331e+00 5.846052e+00 -6.409123e-01 -7.937512e+00 2.799239e-02
[2146] -1.858898e+00 4.399454e+00 -6.480302e-01 -2.130195e+00 -4.382969e+00
[2151] 5.435263e-01 1.654628e-01 1.032985e+01 -5.165389e-01 2.651794e+00
[2156] -1.113397e+00 2.522320e+01 -9.480667e-02 -3.777304e-01 -1.361166e-01
[2161] -6.967938e-01 -7.677110e+00 -4.484681e+00 -7.449997e-01 -9.374488e-01
[2166] -4.122238e-01 9.026965e-01 3.020804e+00 -1.386374e+00 -1.873359e+01
[2171] -1.070408e+00 5.521024e+00 -4.116253e-01 -2.577428e-01 -2.512488e+00
[2176] 7.983223e-01 1.581338e+00 -3.257811e-01 -3.198663e-01 -2.595229e+00
[2181] 6.390141e-01 -8.418747e-02 -3.970085e-01 -1.705716e-01 -1.776100e+01
[2186] -2.175716e+00 -1.126550e+00 8.975584e+00 -4.012066e-01 1.399403e+01
[2191] 1.587782e+01 1.023151e-01 7.059435e-01 4.161542e-01 -4.013470e+00
[2196] 1.189279e-01 -6.673784e-01 -4.198970e-01 1.108788e+00 4.016482e-01
[2201] -9.222420e+00 2.315440e+01 2.131762e+00 -3.374852e-01 -7.923102e+00
[2206] 1.490549e-01 -1.837561e+00 -2.478501e-01 2.260515e+00 2.817259e+00
[2211] 1.417058e+00 1.728184e+00 1.020994e+00 -2.929953e+00 1.419116e-01
[2216] 2.614197e+00 1.390382e+00 1.339087e-01 2.109118e+00 -7.054273e+00
[2221] 2.095708e-01 7.451882e-01 -2.301187e-01 1.631462e+00 1.075897e+00
[2226] 1.437309e+00 2.981027e-01 3.271638e+00 2.340605e-01 -6.274342e-01
[2231] -6.041938e-01 8.411647e-01 7.535063e-02 1.478397e+01 -2.023845e+01
[2236] -1.057925e+00 -3.562701e-01 -2.587477e-01 9.261567e-01 2.544068e-01
[2241] -4.956176e-01 9.569976e-01 5.058740e-01 7.945952e-02 1.426947e+01
[2246] -3.236090e-01 -1.597363e-02 1.568278e+00 2.745446e+00 -6.624559e-01
[2251] -6.620051e-01 6.929064e+00 -3.395485e-01 -7.331020e-01 2.007203e+00
[2256] -2.928672e+00 -1.684684e-01 1.375844e+00 2.454376e+02 7.990356e-01
[2261] -1.027425e+01 -5.857654e-01 -1.900024e+00 -2.195226e+00 2.781591e-01
[2266] 4.038760e-01 -1.544292e-01 4.711869e-02 4.841462e+00 1.031867e-01
[2271] -8.611969e-01 -1.664425e+00 -1.228268e+00 -1.152710e+00 -1.168115e+01
[2276] 5.205492e+00 1.192407e+00 1.200924e+00 5.200826e-01 -2.963393e-01
[2281] 2.231061e+00 3.532329e-01 -4.784655e+00 1.508867e+00 2.391733e+02
[2286] -5.575948e+00 -1.660505e-01 -2.378567e+00 8.412162e-02 3.120898e-01
[2291] -1.642957e-01 -1.859242e+00 2.374683e+01 1.317778e+00 -2.493574e-02
[2296] 1.062495e+00 7.719270e-01 -4.542170e-01 4.122966e+00 -3.845125e-01
[2301] -6.505943e-01 1.153148e+00 3.064804e+00 4.628565e-01 4.365101e-01
[2306] 1.007231e+00 -2.445697e+00 1.205804e+00 3.215264e+00 -1.096839e-02
[2311] -1.120888e+00 -7.122192e-01 -1.702755e+00 -1.361407e+01 1.208343e+00
[2316] -9.282947e-02 9.673931e-01 -2.245762e+00 -2.566566e+00 2.481839e+00
[2321] -4.869585e+00 -1.452603e+00 -7.807805e-01 -8.431710e-01 7.992314e+00
[2326] -1.888179e+00 4.546482e-01 -3.173849e+00 3.002449e-01 7.025764e-01
[2331] -9.136911e-01 -1.102757e+00 -6.582601e-01 3.699767e-01 -1.929952e+00
[2336] -1.477368e-01 -1.014731e-01 2.623496e+00 -5.435010e+00 -1.892819e+00
[2341] -9.874076e-01 -3.913674e-01 -5.734874e-01 2.779174e+00 -3.627486e-01
[2346] -3.115675e-01 3.088604e+00 3.175347e-02 -1.582482e+00 4.001477e-01
[2351] 1.587834e+00 6.483626e-02 -4.194138e+00 3.153073e+00 7.830437e-01
[2356] 3.646169e-02 -9.009853e-02 2.944153e-01 -7.213604e+00 -3.426930e-01
[2361] 3.127621e+00 -9.015495e+02 1.517887e-02 5.475867e-01 5.085444e-01
[2366] -1.696348e-01 -2.515755e-01 4.172328e-01 -1.543427e-01 -9.988567e-01
[2371] -1.220910e+00 -1.888139e+00 5.107525e-01 -1.817646e+00 1.122587e+00
[2376] 5.188007e+00 -1.436607e-01 -7.283602e-01 -2.939507e-01 1.366986e+00
[2381] 2.445594e+00 1.547766e-01 1.835611e+00 -3.802635e+00 6.188904e+00
[2386] -4.445503e+00 -3.810106e-01 -9.348390e-02 4.668712e+00 1.294260e+00
[2391] 8.172335e-01 5.374584e-01 2.432251e-01 9.955533e-02 1.448967e-01
[2396] -3.314081e+00 2.542721e+00 6.983017e-01 -6.458744e-01 -6.176532e-01
[2401] -5.937590e-01 9.409167e-02 3.566087e+00 1.453226e+00 -8.081475e-01
[2406] 4.409328e-01 6.708021e-01 1.577516e-01 6.399901e-01 3.546097e+00
[2411] -8.004703e-01 3.632253e-01 -8.415333e+00 -3.702194e-01 -8.296650e-01
[2416] 3.009139e+00 6.484510e-01 1.094918e+00 -5.071789e-01 6.628444e-01
[2421] 1.000349e+01 1.057433e+00 -5.231171e+00 2.439074e-01 -4.081160e-02
[2426] -1.837006e+00 -2.744675e-02 -1.372985e+00 -2.660858e+00 6.535175e+00
[2431] -6.236143e+00 -2.017299e+00 4.698356e-02 5.169047e-02 8.897124e-01
[2436] 7.957988e+00 -2.984390e+00 9.342397e+00 -9.242789e-01 -5.093230e+00
[2441] -7.295703e-01 -1.466964e+01 -1.925611e+00 2.076944e+01 1.823310e+01
[2446] -1.197977e+01 8.858440e-01 -1.858427e+00 1.029292e+00 -3.152319e+00
[2451] 5.340341e-01 -3.601272e-01 -1.481907e+00 -1.879257e+00 -1.061363e-01
[2456] -2.310645e+00 4.285935e+00 -1.187563e+00 4.890591e-01 -2.759514e-01
[2461] -1.150631e+00 8.266556e-01 -1.362800e+00 1.898852e+00 2.397034e+00
[2466] 1.355404e-01 -2.450776e-01 4.828021e+00 2.277789e+00 4.312064e+01
[2471] -1.784173e+00 -3.974256e-01 -9.660343e-03 5.518524e+00 -2.558096e+00
[2476] -6.516618e-01 -8.158139e-01 6.488007e-01 -1.725979e+00 7.313601e-01
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[2486] -1.098856e-01 -1.983885e-01 1.593636e+00 8.070401e-01 1.265922e+00
[2491] 3.268355e-02 1.799492e+00 -1.603049e+00 -9.831718e-03 1.443522e+00
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[2516] 1.747736e+01 1.941339e-01 4.228383e-01 -4.520854e-01 4.887517e-01
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[3526] 5.363358e-01 -2.796424e-01 -1.306449e+00 -6.349354e-02 6.712229e+00
[3531] -2.361345e+00 -3.207247e-02 1.161234e-01 6.701439e-01 4.736202e+00
[3536] -1.400518e+00 3.095968e-01 2.318106e+01 -9.355460e+00 7.935388e-01
[3541] -2.222384e+00 -8.689649e-01 -1.367753e+00 7.296377e-01 -8.896150e-02
[3546] -1.485118e+00 2.545346e-01 5.817141e-01 9.835350e-01 -2.474080e-01
[3551] 3.676591e+00 -9.512038e-01 1.110174e+01 -3.186304e+00 -3.038580e+01
[3556] -1.652746e-01 1.139456e+00 -5.914430e-01 -7.877249e-01 -2.062669e+00
[3561] 3.159921e-01 -7.268210e-02 1.538497e-01 8.605571e-01 7.414036e-01
[3566] 9.703574e-02 -9.894662e-01 -1.371595e+01 3.446280e-01 -7.905141e+00
[3571] 1.703760e-01 -3.661620e+00 2.748840e+00 2.726982e-01 -2.758299e+00
[3576] 5.012421e-01 -1.951417e-01 1.989792e-01 -2.408155e+00 -4.379631e-01
[3581] -2.351007e-01 -7.463257e-01 -6.303043e-01 -3.514313e+00 2.994087e+00
[3586] 2.303571e+00 1.915269e+00 -1.048907e+00 2.343525e+00 1.955974e-01
[3591] -4.654761e+01 1.489282e+00 -8.590761e-01 4.293108e+00 -6.201211e-01
[3596] 3.561348e-01 -2.439519e-02 7.320877e-01 3.222001e+00 -2.090350e+00
[3601] 7.963314e+00 -1.495370e+00 -5.051503e-01 -5.249255e-01 -4.542426e-01
[3606] -9.078993e-01 -1.574700e+00 1.827011e+00 1.605848e+00 -1.476919e+00
[3611] -2.603402e-01 -8.990229e-01 3.408260e+00 -2.565161e-01 3.054100e-01
[3616] 1.040507e+00 -1.177612e-01 -2.071058e+00 1.816584e+00 8.956200e+00
[3621] 1.617753e+00 -2.172705e+00 -6.833003e-01 9.378811e-01 -1.545935e+00
[3626] 8.411715e-01 -1.234221e+00 2.949170e+00 -4.755474e-01 4.396234e+00
[3631] -3.801291e-01 6.053110e-01 1.474366e+01 5.419029e+00 -1.140540e+00
[3636] 1.464072e+00 1.587495e+00 6.482482e-02 7.268655e-01 -8.415497e-01
[3641] -1.031923e+00 -2.588255e+00 1.846480e+00 3.458723e-01 7.882751e+00
[3646] -5.265448e-01 -7.535303e-01 -4.158971e-01 -1.516699e-01 5.479488e-01
[3651] -6.577216e-02 6.356719e-01 -1.898429e+02 -3.552698e+01 -1.191612e-01
[3656] -2.417650e+00 -7.225493e+00 2.945918e-01 1.578779e-01 -7.763222e-01
[3661] 8.125505e-01 -5.569515e-01 5.070760e-01 4.074785e-01 -1.274851e+00
[3666] -3.500656e+00 -1.212448e-01 3.512139e-01 1.810580e+00 -1.449860e-01
[3671] 1.098947e-01 3.775910e-01 -4.296744e-01 -1.118344e+00 5.593056e-01
[3676] 9.961817e-02 9.372262e-01 -3.858298e+00 1.171551e+01 3.762278e+00
[3681] -1.042404e-01 -2.365304e+00 1.031428e+00 -1.311083e+00 -1.885038e+00
[3686] -2.213389e+00 1.610543e-01 7.366706e+00 2.122968e-01 -3.065079e+00
[3691] 1.024416e+01 2.500711e-02 -3.080425e+00 -6.210310e-01 4.204067e-01
[3696] 1.710029e+00 -3.458445e+00 7.505376e+00 5.373030e-01 5.586223e+00
[3701] 1.244446e-01 -1.384025e+00 -9.809359e+00 -4.867494e-02 2.267091e-01
[3706] 1.207749e+01 6.023497e-01 -2.997571e+00 1.287506e+00 -2.519516e-01
[3711] -8.617084e+00 1.926113e+00 -3.306969e+00 -1.721307e+00 -1.513541e+00
[3716] 4.227904e-01 2.126175e+00 1.714784e+02 -1.299363e+00 -4.690841e-01
[3721] 2.460020e+00 -2.686629e+00 -4.430320e-01 -1.045494e-01 -3.973288e-01
[3726] -5.478736e+01 3.335980e+00 -9.792836e-01 2.033550e-01 -5.558787e+00
[3731] -5.464843e-01 -3.748487e-01 6.248667e+00 -2.229574e+00 8.784275e+00
[3736] -1.201689e+00 -2.455076e-02 5.164359e-01 7.189235e+00 1.435354e+00
[3741] -9.908312e-01 -6.405897e-01 -1.701228e+02 -7.965780e+00 -1.538686e+00
[3746] -9.331105e+00 1.043072e+00 3.304424e-01 -2.974133e+00 -1.332391e+00
[3751] -1.085245e+00 3.310437e+00 -2.642783e-02 -5.557477e-01 -3.292065e-01
[3756] -1.064761e+00 1.134183e+00 1.034589e+00 -4.002225e+00 -1.187685e+00
[3761] 2.769520e+00 -1.479536e+00 1.180211e+00 -9.028214e-01 4.725802e-01
[3766] 1.641531e+00 -2.705758e+00 -1.555030e+01 -3.993281e+00 9.684113e+00
[3771] -6.971772e-01 -2.859899e+00 -3.364934e+00 -5.335719e-03 2.970540e+00
[3776] 8.481290e+00 6.337802e-02 1.881617e+00 7.088344e+00 1.185774e-01
[3781] 4.016650e+00 -2.404269e-01 1.983642e+00 1.270141e+00 -3.751150e+00
[3786] 2.674283e+00 8.876443e+00 4.525067e-01 -5.231408e-01 1.585257e-01
[3791] 3.697809e+01 2.251768e+00 -1.102302e+00 1.117070e-01 -1.053623e-01
[3796] -1.100615e+00 6.338987e-01 2.478740e-01 1.024970e+00 -2.633222e+00
[3801] -4.009569e+00 -1.101052e+00 -4.060655e-01 3.004637e+00 5.814043e-01
[3806] -6.327610e-02 3.325765e-01 2.064858e+00 -2.145155e+00 7.460232e-01
[3811] 5.290967e+00 1.528458e+01 -6.205097e-01 -9.838407e+01 8.461809e+00
[3816] 6.677553e+00 -1.820412e+00 2.157305e-01 3.334248e+00 -2.530015e-02
[3821] 9.525268e-02 -5.161167e+01 -6.152761e-01 7.416282e-01 2.635613e-01
[3826] 5.437256e-01 -2.380667e+00 3.974642e-02 -1.728096e+00 4.071759e-01
[3831] 2.690098e-01 1.591545e+00 -2.775746e+00 -8.718777e-01 1.110227e+00
[3836] -4.071743e-01 3.461507e+00 7.713410e-01 -2.973885e+00 -1.573023e-01
[3841] 1.234028e-01 6.264572e-01 9.885863e-01 -3.327504e-01 -7.345798e-01
[3846] 3.497987e-01 6.711337e-01 5.861236e+00 -3.196129e-01 3.082762e-01
[3851] -1.922805e+00 1.025224e-01 -3.521623e-01 2.144386e-01 1.817203e+01
[3856] 4.077217e-01 -1.285562e+00 -3.985437e-01 3.109252e+00 1.656315e-01
[3861] -1.286470e+00 -1.837142e+00 8.029093e-01 4.744065e-01 -7.949851e+00
[3866] -1.717412e+00 -4.962705e+00 -1.379434e+00 -5.257068e-02 6.861902e-01
[3871] -4.647929e+00 2.089369e+00 1.304218e+01 -6.226730e-01 1.512918e-01
[3876] -1.486341e+00 3.742063e+00 -1.213660e+00 2.417520e+00 1.152834e-01
[3881] 3.358816e+00 1.060214e+00 -1.588688e+00 -6.833458e-01 1.389630e+00
[3886] -9.266602e-01 6.447597e-01 -1.681534e+00 1.769815e+01 2.035199e+00
[3891] 2.939207e-01 6.041787e+00 -1.288368e+02 -2.173490e+00 -4.036653e-01
[3896] -2.958568e-02 -7.446158e-01 -1.571518e-02 2.398758e+00 5.684703e+01
[3901] -1.685815e+00 -1.355883e+00 -4.635198e-01 -3.324906e-01 1.085364e+00
[3906] 3.646224e+00 1.213343e+00 -1.124866e-01 -1.051977e+00 -1.289229e+00
[3911] 1.042275e+00 -1.177983e+00 -3.311629e+00 1.009457e+00 -4.567665e-01
[3916] -1.090413e+00 1.302620e+01 -3.756236e-01 -1.728149e+00 -6.126915e+00
[3921] -1.892961e+00 6.088260e-01 7.672290e-02 7.027673e-01 -2.258337e+00
[3926] -1.050254e+00 6.923946e-01 -3.168746e+00 8.096458e-01 2.653325e+00
[3931] 1.704836e+00 9.847437e-01 -3.465838e-01 -3.183693e-01 8.118799e+00
[3936] -3.091133e+00 -4.307849e-01 2.294197e-01 5.804242e-03 -8.115063e-01
[3941] -6.565940e-01 -1.141588e+00 -3.229612e-01 -4.579888e-01 -1.297318e+00
[3946] -5.584553e-01 -1.311275e+00 3.552158e-01 2.614657e+01 7.671520e-01
[3951] -1.023908e+00 -2.987707e-01 4.520979e+00 -1.510258e+00 1.056455e+00
[3956] 4.857066e-02 -2.303134e-01 8.620867e-01 -1.246282e+00 1.459209e+00
[3961] -8.286817e+00 -1.646985e+00 4.307915e+00 7.109435e-01 -1.168944e+00
[3966] -2.485401e+01 5.878702e-01 1.044863e-01 7.548677e+00 -5.818995e+00
[3971] 5.026403e+00 2.140157e+00 -4.031617e-01 -4.068551e-01 6.024553e-01
[3976] 6.640828e-03 1.396460e+00 -8.827639e-01 -2.049025e+00 6.400908e-01
[3981] -1.903370e+00 4.245265e-01 -6.115154e-01 -9.950276e-01 -4.869529e-01
[3986] 1.210167e-01 1.376209e+00 2.940083e-01 -1.630650e+00 -2.053272e+01
[3991] 1.652995e+00 -5.975635e-01 -5.587384e+00 6.830938e-01 1.806467e+01
[3996] 4.212418e-01 -1.124877e+00 4.628983e+00 6.713240e-01 -3.570027e-01
(c) Usando las funciones de la librería patchwork ponga en un mismo gráfico las cuatro
gráficas de las funciones de masa del inicio (a).
eliminar_outliers<-function(data){
# Calculamos el primer cuartil (Q1) y el tercer cuartil (Q3)
Q1 <- quantile(data, 0.25)
Q3 <- quantile(data, 0.75)
# Calculamos el rango intercuartil (IQR)
IQR <- Q3 - Q1
# Establecemos límites para detectar outliers
lower_bound <- Q1 - 1.5 * IQR
upper_bound <- Q3 + 1.5 * IQR
# Identificamos y eliminamos outliers
return(data[data > lower_bound & data < upper_bound])
}
crear_histograma <- function(datos, titulo) {
df <- data.frame(x = datos)
ggplot(df, aes(x)) +
geom_histogram(binwidth = 1, color = "black", fill = "lightblue", boundary = 0.5) +
scale_x_continuous(breaks = seq(min(datos), max(datos), 1)) +
labs(title = titulo, x = "(X)", y = "Frecuencia") +
theme_minimal()
}
# Creamos histogramas (3 por la tardanza)
hist_n100 <- crear_histograma(eliminar_outliers(med_arit[1:1000]), "Histograma de n = 100")
hist_n1000 <- crear_histograma(eliminar_outliers(med_arit[1001:2000]), "Histograma de n = 1000")
hist_n10000 <- crear_histograma(eliminar_outliers(med_arit[2001:3000]), "Histograma de n = 10000")
hist_n100000 <- crear_histograma(eliminar_outliers(med_arit[3001:4000]), "Histograma de n = 100000")
# Organizamos los histogramas en una matriz 2x2 utilizando patchwork
(hist_n100 | hist_n1000) / (hist_n10000 | hist_n100000)
(d) ¿Qué densidad dirÍa que tiene \(\bar{X}\)?
Parece una normal con \(media=0\)
(e) ¿DirÍa que se violenta el Teorema del LÍmite Central?
No se violenta, dado que se trata de una Cauchy, sabemos que su media es indeterminada
14. Responda las siguientes preguntas:
(a) Considere el lanzamiento de 2 dados y sea \(X\) la suma sus valores. Lleve a cabo 100,000
simulaciones y obtenga una aproximación de la función de masa de \(X\)
dados_simulacion <- function(n) {
dados <- 1:6
resultados <- numeric(n)
for (i in 1:n) {
muestra <- sample(dados, 2, replace = FALSE)
suma <- sum(muestra)
resultados[i] <- suma
}
return(resultados)
}
sim100000<-dados_simulacion(100000)crear_histograma <- function(datos, titulo) {
df <- data.frame(x = datos)
ggplot(df, aes(x)) +
geom_histogram(binwidth = 1, color = "black", fill = "lightblue", boundary = 0.5) +
scale_x_continuous(breaks = seq(min(datos), max(datos), 1)) +
labs(title = titulo, x = "(X)", y = "Frecuencia") +
theme_minimal()
}
# Crear histogramas
hist_n100 <- crear_histograma(sim100000, "Histograma de n = 100")
hist_n100
(b) 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 \(Y\)
hexa_simulacion <- function(n) {
dados <- c(2,2,2,6,6,6)
resultados <- numeric(n)
for (i in 1:n) {
muestra <- sample(dados, 1, replace = FALSE)
suma <- 5+muestra
resultados[i] <- suma
}
return(resultados)
}
sim_hex100000<-hexa_simulacion(100000)
crear_histograma <- function(datos, titulo) {
df <- data.frame(x = datos)
ggplot(df, aes(x)) +
geom_histogram(binwidth = 1, color = "black", fill = "lightblue", boundary = 0.5) +
scale_x_continuous(breaks = seq(min(datos), max(datos), 1)) +
labs(title = titulo, x = "(X)", y = "Frecuencia") +
theme_minimal()
}
# Creamos histogramas
hist_hex <- crear_histograma(sim_hex100000, "Histograma de n = 100")
hist_hex
(c) 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 de masa de \(Z\).
hexa2_simulacion <- function(n) {
dados <- c(1,2,2,3,3,4)
dados2 <- c(1,3,4,5,6,8)
resultados <- numeric(n)
for (i in 1:n) {
muestra <- sample(dados, 1, replace = FALSE)
muestra2 <- sample(dados2, 1, replace = FALSE)
suma <- muestra2+muestra
resultados[i] <- suma
}
return(resultados)
}
sim_hex100000<-hexa2_simulacion(100000)
crear_histograma <- function(datos, titulo) {
df <- data.frame(x = datos)
ggplot(df, aes(x)) +
geom_histogram(binwidth = 1, color = "black", fill = "lightblue", boundary = 0.5) +
scale_x_continuous(breaks = seq(min(datos), max(datos), 1)) +
labs(title = titulo, x = "(X)", y = "Frecuencia") +
theme_minimal()
}
# Creamos histogramas
hist_hex <- crear_histograma(sim_hex100000, "Histograma de n = 100")
hist_hex
(d) ¿Qué puede decir de las densidades de \(X\), \(Y\) y \(Z\)?
La primera no tiene pinta de una distribución que yo conozca.
La segunda se distribuye bernoulli soporte \((7,11)\) con \(p=.5\)
La tercera parece una normal centrada en 7