📊 Tarea Capitulo 3

📅 21 de Octubre de 2025

⚽

Nikole Gutierrez
💙 Ingeniería Mecánica

🥅

Antonio Garcia
💙 Ingeniería Civil

👟

Luisa De Angel
💙 Ingeniería Civil

library(flextable)
library(dplyr)
library(MASS)
library(ggplot2)

Ejercicio 3.7

# Definir la función de densidad
f <- function(x) {
  ifelse(x > 0 & x < 1, x,
         ifelse(x >= 1 & x < 2, 2 - x, 0))}


# Función de distribución acumulada F(x)
F <- function(x) {
  ifelse(x < 0, 0,
         ifelse(x < 1, (x^2) / 2,
                ifelse(x < 2, 2*x - (x^2)/2 - 1, 1)))} #Función integrada

total_area <- integrate(f, lower = 0, upper = 2)$value
cat("Área total =", total_area, "\n")

## Área total = 1

a) Menos de 120 horas

# P(X < 1.2), como es sobre 100 horas, el porcentaje de 120 horas será 1.2
prob_a <- F(1.2)
cat("P(X < 1.2) =", prob_a, "\n") 

## P(X < 1.2) = 0.68

b) Entre 50 y 100 horas

# P(0.5 ≤ X ≤ 1)
prob_b <- F(1) - F(0.5)
cat("P(0.5 ≤ X ≤ 1) =", prob_b, "\n")  

## P(0.5 ≤ X ≤ 1) = 0.375

Gráfica de densidad

library(ggplot2)
x <- seq(-0.5, 2.5, by = 0.001)
tb <- data.frame(x = x, y = f(x))

ggplot(tb, aes(x = x, y = y)) +
  geom_line(color = "black", size = 1.2) +
  geom_area(data = subset(tb, x >= 0 & x <= 1.2),
            aes(x = x, y = y), fill = "lightblue", alpha = 0.4) +
  geom_area(data = subset(tb, x >= 0.5 & x <= 1),
            aes(x = x, y = y), fill = "blue", alpha = 0.5) +
  labs(title = "Funcion de densidad f(x)",
     subtitle = "Areas sombreadas: P(X<1.2) morado, P(0.5≤X≤1) naranja",
     x = "x (centenas de horas)", y = "f(x)") +
  theme_bw()

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

Gráfica de la función de distribución acumulada

x <- seq(-0.5, 2.5, by = 0.001)
tb_F <- data.frame(x = x, y = F(x))

ggplot(tb_F, aes(x = x, y = y)) +
  geom_line(color = "darkblue", size = 1.2) +
  geom_point(aes(x = 1.2, y = F(1.2)), color = "red", size = 3) +
  geom_hline(yintercept = F(1.2), linetype = "dashed", color = "gray") +
  labs(title = "Funcion de distribucion acumulada F(x)",
     subtitle = "P(X < 1.2) = 0.68",
     x = "x (centenas de horas)", y = "F(x)") +
  theme_bw()

## Warning in geom_point(aes(x = 1.2, y = F(1.2)), color = "red", size = 3): All aesthetics have length 1, but the data has 3001 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.

Ejercico 3.9

f.x <- function(x){
  ifelse(x > 0 & x < 1, 2*(x + 2)/5, 0)}

F.x <- function(x){
  ifelse(x <= 0, 0,
         ifelse(x < 1, (x^2 + 4*x)/5, 1))} #Función integrada

a) Demuestre que P(0 < X < 1) = 1

total_area <- integrate(f.x, lower = 0, upper = 1)$value
cat("Integral total (0,1) =", total_area, "\n")

## Integral total (0,1) = 1

b) Calcule la probabilidad de que más de 1/4 pero menos de 1/2 de las personas contactadas respondan a este tipo de encuesta

p_a <- 1  
p_b <- integrate(f.x, lower = 1/4, upper = 1/2)$value
cat("P(1/4 < X < 1/2) =", p_b, "   (fraccion = 19/80 = 0.2375)\n")

## P(1/4 < X < 1/2) = 0.2375    (fraccion = 19/80 = 0.2375)

Gráfica de densidad

# Crear tabla para graficas
x <- seq(-0.2, 1.2, by = 0.001)
df <- data.frame(x = x, pdf = f.x(x), cdf = F.x(x))

# Grafica con area sombreada entre 1/4 y 1/2
df_area_b <- subset(df, x >= 1/4 & x <= 1/2)

g1 <- ggplot(df, aes(x = x, y = pdf)) +
  geom_line(size = 1.1, color = "black") +
  geom_area(data = subset(df, x >= 0 & x <= 1), aes(x=x, y=pdf), fill = "grey90") +
  geom_area(data = df_area_b, aes(x = x, y = pdf), fill = "blue", alpha = 0.6) +
  labs(title = "Funcion de densidad f(x)",
       subtitle = "Area azul = P(1/4 < X < 1/2)",
       x = "x (proporcion, 0-1)", y = "f(x)") +
  theme_bw()
print(g1)

Gráfica de distribución

# Grafica de distribución
g2 <- ggplot(df, aes(x = x, y = cdf)) +
  geom_line(size = 1.1, color = "blue") +
  geom_vline(xintercept = c(1/4, 1/2), linetype = "dashed") +
  geom_point(data = data.frame(x=c(1/4,1/2), y=c(F.x(1/4), F.x(1/2))),
             aes(x=x,y=y), color="red", size=2) +
  labs(title = "Funcion de distribucion acumulada F(x)",
       x = "x (proporcion, 0-1)", y = "F(x)") +
  theme_bw()
print(g2)

Ejercicio 3.13

\(\small X\) \[ \begin{array}{c|c} x & f(x) \\ \hline 0 & 0.41 \\ 1 & 0.37 \\ 2 & 0.16 \\ 3 & 0.05 \\ 4 & 0.01 \\ \end{array} \]

Función de distribución acumulativa

\[ F(X) = \begin{cases} 0 & \text{si } x < 0, \\ 0.41 & \text{si } 0 \le x < 1, \\ 0.78 & \text{si } 1 \le x < 2, \\ 0.94 & \text{si } 2 \le x < 3, \\ 0.99 & \text{si } 3 \le x < 4, \\ 1 & \text{si } x \ge 4 \end{cases} \]

Construcción de la función de distribución:

F.x <- function(x){
  ifelse(x<0,0,
         ifelse(x<1,0.41,
                ifelse(x < 2, 0.78,
                       ifelse(x < 3,0.94,
                              ifelse(x<4,0.99,1)))))
}

F.x(2.8)

## [1] 0.94

Ejercicio 3.12

library(flextable)
library(ggplot2)

# Definir la función f(t)
F_t <- function(t) {
ifelse(t < 1, 0,
ifelse(t < 3, 1/4,
ifelse(t < 5, 1/2,
ifelse(t < 7, 3/4, 1))))}

# Valores posibles de T y sus probabilidades puntuales
valores_T <- c(1, 3, 5, 7)
prob_T <- rep(1/4, 4)


tabla_T <- data.frame(
"T (años)" = valores_T,
"P(T=t)" = prob_T
)


ft_tabla_T <- flextable(tabla_T)
ft_tabla_T <- set_header_labels(ft_tabla_T,
"T (años)" = "T (años)",
"P(T=t)" = "P(T = t)")
ft_tabla_T <- theme_vanilla(ft_tabla_T)
ft_tabla_T <- autofit(ft_tabla_T)
ft_tabla_T <- color(ft_tabla_T, color = "black", part = "all")
ft_tabla_T <- bg(ft_tabla_T, bg = "#e6f0ff", part = "body")
ft_tabla_T <- bold(ft_tabla_T, part = "header")
ft_tabla_T

T..años.	P.T.t.
1	0.25
3	0.25
5	0.25
7	0.25

Grafica de F(t)

t_vals <- seq(0, 8, 0.01)
df_F <- data.frame(t = t_vals, F = F_t(t_vals))

ggplot(df_F, aes(x = t, y = F)) +
geom_segment(aes(x = 1, xend = 3, y = 0.25, yend = 0.25), color = "blue", size = 1.3) +
geom_segment(aes(x = 3, xend = 5, y = 0.5, yend = 0.5), color = "blue", size = 1.3) +
geom_segment(aes(x = 5, xend = 7, y = 0.75, yend = 0.75), color = "blue", size = 1.3) +
geom_segment(aes(x = 7, xend = 8, y = 1, yend = 1), color = "blue", size = 1.3) +
geom_point(data = data.frame(t = valores_T, F = F_t(valores_T)),
aes(x = t, y = F), color = "red", size = 3) +
labs(title = "Función de distribución acumulativa F(t)",
subtitle = "Ejercicio 3.12 — Bonos municipales",
x = "t (años)", y = "F(t)") +
theme_bw() +
theme(plot.title = element_text(face = "bold"))

## Warning in geom_segment(aes(x = 1, xend = 3, y = 0.25, yend = 0.25), color = "blue", : All aesthetics have length 1, but the data has 801 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.

## Warning in geom_segment(aes(x = 3, xend = 5, y = 0.5, yend = 0.5), color = "blue", : All aesthetics have length 1, but the data has 801 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.

## Warning in geom_segment(aes(x = 5, xend = 7, y = 0.75, yend = 0.75), color = "blue", : All aesthetics have length 1, but the data has 801 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.

## Warning in geom_segment(aes(x = 7, xend = 8, y = 1, yend = 1), color = "blue", : All aesthetics have length 1, but the data has 801 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.

Ejercicio 3.21

# Definir la función de densidad f(x)
f <- function(x) {
  k <- 3/2  # Valor de k calculado
  ifelse(x > 0 & x < 1, k * sqrt(x), 0)}


# Definir la función de distribución acumulada F(x)

F <- function(x) {
  ifelse(x <= 0, 0,
         ifelse(x < 1, x^(3/2), 1))}


total_area <- integrate(f, lower = 0, upper = 1)$value
cat("Área total =", total_area, "\n")

## Área total = 1

a) Evaluar k

integral_valor <- integrate(function(x) sqrt(x), lower = 0, upper = 1)$value
k <- 1 / integral_valor
cat("El valor de k que hace que f(x) sea una densidad es: k =", k, "\n")

## El valor de k que hace que f(x) sea una densidad es: k = 1.5

b) Calcule F(x) y utilice el resultado para evaluar

P(0.3 < X < 0.6).

# Definir nuevamente la función f(x) con el valor de k = 1.5
f <- function(x) {
  ifelse(x > 0 & x < 1, 1.5 * sqrt(x), 0)}

# Calcular P(0.3 < X < 0.6)
prob_b <- integrate(f, lower = 0.3, upper = 0.6)$value
cat("P(0.3 < X < 0.6) =", prob_b, "\n")

## P(0.3 < X < 0.6) = 0.3004412

# Validar que el área total bajo f(x) sea 1
total_area <- integrate(f, lower = 0, upper = 1)$value
cat("Área total =", total_area, "\n")

## Área total = 1

Gafica de densidad

library(ggplot2)
x <- seq(-0.2, 1.2, by = 0.001)
tb <- data.frame(x = x, y = f(x))

## Warning in sqrt(x): Se han producido NaNs

ggplot(tb, aes(x = x, y = y)) +
  geom_line(color = "black", size = 1.2) +
  geom_area(data = subset(tb, x >= 0.3 & x <= 0.6),
            aes(x = x, y = y), fill = "lightblue", alpha = 0.5) +
  labs(title = "Ejercicio 3.21 - f(x) = 1.5√x",
       subtitle = "Área azul = P(0.3 < X < 0.6)",
       x = "x", y = "f(x)") +
  theme_bw()

Grafica de distribucion acumulada

x <- seq(-0.2, 1.2, by = 0.001)
tb_F <- data.frame(x = x, y = F(x))

ggplot(tb_F, aes(x = x, y = y)) +
  geom_line(color = "darkblue", size = 1.2) +
  geom_point(aes(x = 0.3, y = F(0.3)), color = "red", size = 3) +
  geom_point(aes(x = 0.6, y = F(0.6)), color = "red", size = 3) +
  geom_segment(aes(x = 0.3, y = 0, xend = 0.3, yend = F(0.3)), linetype="dashed", color="gray") +
  geom_segment(aes(x = 0.6, y = 0, xend = 0.6, yend = F(0.6)), linetype="dashed", color="gray") +
  labs(title = "Ejercicio 3.21 - Función de distribución F(x)",
       subtitle = "P(0.3 < X < 0.6) = F(0.6) - F(0.3)",
       x = "x", y = "F(x)") +
  theme_bw()

## Warning in geom_point(aes(x = 0.3, y = F(0.3)), color = "red", size = 3): All aesthetics have length 1, but the data has 1401 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.

## Warning in geom_point(aes(x = 0.6, y = F(0.6)), color = "red", size = 3): All aesthetics have length 1, but the data has 1401 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.

## Warning in geom_segment(aes(x = 0.3, y = 0, xend = 0.3, yend = F(0.3)), : All aesthetics have length 1, but the data has 1401 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.

## Warning in geom_segment(aes(x = 0.6, y = 0, xend = 0.6, yend = F(0.6)), : All aesthetics have length 1, but the data has 1401 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.

Ejercicio 3.27

# función de densidad f(x)
f.x <- function(x) {
  ifelse(x >= 0, (1/2000) * exp(-x / 2000), 0)
}

a) Calcular F(x) por integración

creamos una función que integra f.x desde 0 hasta x (para x >= 0)

library(flextable)

# ---- Cálculo de F(x) por integral ----
F_from_integral <- function(x) {
  sapply(x, function(t) {
    if (t < 0) return(0)
    integrate(f.x, lower = 0, upper = t)$value
  })
}

# Puntos a evaluar
x_eval <- c(0, 500, 1000, 2000, 3000)
F_integral_vals <- F_from_integral(x_eval)

# ---- Cálculo analítico ----
F_analytic <- function(x) ifelse(x < 0, 0, 1 - exp(-x / 2000))
F_analytic_vals <- F_analytic(x_eval)

# ---- Tabla comparativa ----
tabla_F <- data.frame(
  "x (horas)" = x_eval,
  "F(x) por integral" = round(F_integral_vals, 6),
  "F(x) analítica" = round(F_analytic_vals, 6),
  "Diferencia" = round(abs(F_integral_vals - F_analytic_vals), 6)
)

# ---- Mostrar con formato ----
flextable(tabla_F) |>
  set_caption("Comparación entre F(x) por integración numérica y forma analítica") |>
  theme_vanilla() |>
  color(color = "black", part = "all") |>
  bg(bg = "#e6f0ff", part = "header") |>
  align(align = "center", part = "all") |>
  autofit()

Comparación entre F(x) por integración numérica y forma analítica

x..horas.	F.x..por.integral	F.x..analítica	Diferencia
0	0.000000	0.000000	0
500	0.221199	0.221199	0
1,000	0.393469	0.393469	0
2,000	0.632121	0.632121	0
3,000	0.776870	0.776870	0

b) Probabilidad P(X > 1000)

Usando la forma analítica: P(X > 1000) = 1 - F(1000) = exp(-1000/2000)

P_mas_1000 <- 1 - F_analytic(1000)
cat("P(X > 1000) =", round(P_mas_1000, 6), " (≈", round(P_mas_1000*100,2), "% )\n")

## P(X > 1000) = 0.606531  (≈ 60.65 % )

c) Probabilidad P(X < 2000)

P_menor_2000 <- F_analytic(2000)
cat("P(X < 2000) =", round(P_menor_2000, 6), " (≈", round(P_menor_2000*100,2), "% )\n")

## P(X < 2000) = 0.632121  (≈ 63.21 % )

# Ejemplo extra: P(1000 < X < 3000)
P_entre_1000_3000 <- F_analytic(3000) - F_analytic(1000)
cat("P(1000 < X < 3000) =", round(P_entre_1000_3000, 6), "\n\n")

## P(1000 < X < 3000) = 0.3834

Gráfica de densidad (con áreas sombreadas para P(X<2000) y P(1000<X<3000))

library(ggplot2)

x_seq <- seq(0, 8000, by = 1)
df_den <- data.frame(x = x_seq, y = f.x(x_seq))

p_den <- ggplot(df_den, aes(x = x, y = y)) +
  geom_line(color = "black", size = 1) +
  geom_area(data = subset(df_den, x >= 0 & x <= 2000), aes(x = x, y = y),
            fill = "lightblue", alpha = 0.4) +
  geom_area(data = subset(df_den, x >= 1000 & x <= 3000), aes(x = x, y = y),
            fill = "steelblue", alpha = 0.35) +
  labs(title = "Ejercicio 3.27 — Densidad f(x) = (1/2000)e^{-x/2000}",
       subtitle = "Área claro = P(X < 2000). Área oscuro = P(1000 < X < 3000)",
       x = "Horas", y = "f(x)") +
  theme_bw()

print(p_den)

Gráfica de la función de distribución acumulada F(x) (analítica)

df_cdf <- data.frame(x = x_seq, y = F_analytic(x_seq))

p_cdf <- ggplot(df_cdf, aes(x = x, y = y)) +
  geom_line(color = "darkblue", size = 1) +
  geom_vline(xintercept = c(1000, 2000, 3000), linetype = "dashed", color = "gray40") +
  geom_point(aes(x = 1000, y = F_analytic(1000)), color = "red", size = 2) +
  geom_point(aes(x = 2000, y = F_analytic(2000)), color = "red", size = 2) +
  geom_point(aes(x = 3000, y = F_analytic(3000)), color = "red", size = 2) +
  annotate("text", x = 1000, y = F_analytic(1000)+0.03,
           label = paste0("F(1000)=", round(F_analytic(1000),3)), size = 3) +
  annotate("text", x = 2000, y = F_analytic(2000)+0.03,
           label = paste0("F(2000)=", round(F_analytic(2000),3)), size = 3) +
  annotate("text", x = 3000, y = F_analytic(3000)+0.03,
           label = paste0("F(3000)=", round(F_analytic(3000),3)), size = 3) +
  labs(title = "Ejercicio 3.27 — Función de distribución acumulada F(x)",
       x = "Horas", y = "F(x)") +
  theme_bw()

print(p_cdf)

## Warning in geom_point(aes(x = 1000, y = F_analytic(1000)), color = "red", : All aesthetics have length 1, but the data has 8001 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.

## Warning in geom_point(aes(x = 2000, y = F_analytic(2000)), color = "red", : All aesthetics have length 1, but the data has 8001 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.

## Warning in geom_point(aes(x = 3000, y = F_analytic(3000)), color = "red", : All aesthetics have length 1, but the data has 8001 rows.
## ℹ Please consider using `annotate()` or provide this layer with data containing
##   a single row.