TALLERES EN R - RESPUESTAS


Aclaración: Este taller lo hicimos con ayuda del monitor, le agradeceos por eso.


TALLER 322


Problema 1

# Probabilidades 
punif(12,10,20)
## [1] 0.2
punif(15,10,20)-punif (13,10,20)
## [1] 0.2
1-punif(18,10,20)
## [1] 0.2
# Esperanza y Varianza

a <- 10
b <- 20

EX <- (a + b)/2 
VX <- (b - a)^2 / 12
EX;VX
## [1] 15
## [1] 8.333333
x <- seq(5, 25, length = 1000)
y <- dunif(x, min = a, max = b)


plot(x, y, type = "l", xlab = "X", ylab = "Densidad")
polygon(c(10, 10:20, 20),                      # bordes y area 
        c(0 , rep(1/(b-a), length(10:20)), 0), # fx(x)
        col = "lightblue")

Problema 2

pnorm(0.85)                       # (a) P(X > -0.85) = P(X < 0.85)
## [1] 0.8023375
1- pnorm(-0.85)                   # (a) P(X > -0.85) = P(X < -0.85)
## [1] 0.8023375
pnorm( 1.30) - pnorm(0.40)        # (b) P(a < X < b) = F(b) - F(a)
## [1] 0.2477778
pnorm(0.90) - pnorm(0.30)         # (c) 
## [1] 0.1980285
pnorm(0.90) - (1-pnorm(-0.30))    # (c) 
## [1] 0.1980285
pnorm(1.5)  - pnorm(0.45)         # (d)
## [1] 0.259548
(1-pnorm(-1.5))- (1-pnorm(-0.45)) # (d)
## [1] 0.259548
# (+) Grafica 
x <- seq(-3, 3, length = 1000) 
y <- dnorm(x)                  # Valores Normales

par(mfrow = c(2, 2), mar = c(3, 3, 2, 1))

# (a)
plot(x, y, type = "l", ylab = "Densidad", xlab = "Z")
abline(v = -0.85, lwd = 1.5)
polygon(c(-0.85, x[x >= -0.85], 3),
        c(0    , y[x >= -0.85], 0), col = "lightblue")
title(main = "a")

# (b)
plot(x, y, type = "l", ylab = "Densidad", xlab = "Z")
abline(v = c(0.40, 1.30), lwd = 2)
polygon(c(0.40, x[x >= 0.40 & x <= 1.30], 1.30),
        c(0   , y[x >= 0.40 & x <= 1.30], 0),
        col = "lightblue")
title(main = "b")

# (c)
plot(x, y, type = "l", ylab = "Densidad", xlab = "Z")
abline(v = c(-0.30, 0.90), lwd = 2)
polygon(c(-0.30, x[x >= -0.30 & x <= 0.90], 0.90),
        c(0    , y[x >= -0.30 & x <= 0.90], 0),
        col = "lightblue")
title(main = "c")

# (d)
plot(x, y, type = "l", ylab = "Densidad", xlab = "Z")
abline(v = c(-1.50, -0.450), lwd = 2)
polygon(c(-1.50, x[x >= -1.50 & x <= -0.45], -0.45),
        c(0    , y[x >= -1.50 & x <= -0.45], 0),
        col = "lightblue")
title(main = "d")

Problema 3

pnorm(0.56)                        # (a)
## [1] 0.7122603
(1-pnorm(-2.93))- (1-pnorm(-2.06)) # (b)
## [1] 0.01800446
pnorm(0.70) - pnorm(-1.08)         # (c)
## [1] 0.6179653
pnorm(1.62) - pnorm(0.96)          # (d)
## [1] 0.1159115
# (+) Grafica 
x <- seq(-3, 3, length = 1000) 
y <- dnorm(x)                  # Valores Normales

par(mfrow = c(2, 2), mar = c(3, 3, 2, 1))

# (a)
plot(x, y, type = "l", ylab = "Densidad", xlab = "Z")
abline(v = 0.56, lwd = 1.5)
polygon(c(0.56, x[x >= 0.56], 3),
        c(0    , y[x >= 0.56], 0), col = "lightblue")
title(main = "a")

# (b)
plot(x, y, type = "l", ylab = "Densidad", xlab = "Z")
abline(v = c(-2.93, -2.06), lwd = 2)
polygon(c(-2.93, x[x >= -2.93 & x <= -2.06], -2.06),
        c(0   , y[x >= -2.93 & x <= -2.06], 0),
        col = "lightblue")
title(main = "b")

# (c)
plot(x, y, type = "l", ylab = "Densidad", xlab = "Z")
abline(v = c(-1.08, 0.70), lwd = 2)
polygon(c(-1.08, x[x >= -1.08 & x <= 0.70], 0.70),
        c(0    , y[x >= -1.08 & x <= 0.70], 0),
        col = "lightblue")
title(main = "c")

# (d)
plot(x, y, type = "l", ylab = "Densidad", xlab = "Z")
abline(v = c(0.96, 1.62), lwd = 2)
polygon(c(0.96, x[x >= 0.96 & x <= 1.62], 1.62),
        c(0    , y[x >= 0.96 & x <= 1.62], 0),
        col = "lightblue")
title(main = "d")

Problema 4

#a
mu    <- 150
sigma <- sqrt(1000)
x     <- seq(mu - 3*sigma, mu + 3*sigma, length = 1000)
y     <- dnorm(x, mean = mu, sd = sigma)

plot(x, y, type = "l", xlab = "X", ylab = "Densidad")
polygon(c(x, rev(x)), c(y, rep(0, length(y))),
        col = "lightblue", border = "black")

# (b)
pnorm(100, mean = mu, sd = sigma)             # (1) 
## [1] 0.05692315
pnorm(400, mu, sigma) - pnorm(300, mu, sigma) # (2) 
## [1] 1.050718e-06
k1 <- qnorm(0.95)                             # (3) 
k2 <- qnorm(0.975)                            # (4) 

k_90 <- k1 * sigma
k_95 <- k2 * sigma

Problema 5

lambda <- 2
pexp(0, rate = lambda)     # (a)
## [1] 0
1- pexp(2, rate = lambda)  # (b)
## [1] 0.01831564
pexp(2, rate = lambda) - pexp(1, rate = lambda) # (c)
## [1] 0.1170196
# (d)
x      <- seq(0, 5, length = 1000)
y      <- dexp(x, rate = lambda)

plot(x, y, type = "l")
polygon(c(x, rev(x)), c(y, rep(0, length(y))),
        col = "lightblue", border = NA)

## Problema 6

# Definir media y desviación estándar
media <- 480
desv <- 90

# 1. Proporción de puntuaciones mayores a 700
1 - pnorm(700, mean = media, sd = desv)
## [1] 0.007253771
# 2. Percentil 25 de las puntuaciones
qnorm(0.25, mean = media, sd = desv)
## [1] 419.2959
# 3. Si alguien obtuvo 600, ¿en qué percentil está?
pnorm(600, mean = media, sd = desv)
## [1] 0.9087888
# 4. Proporción de puntuaciones entre 420 y 520
pnorm(520, mean = media, sd = desv) - pnorm(420, mean = media, sd = desv)
## [1] 0.4191468

Problema 7

# Definir media y desviación estándar
media <- 170.6
desv <- 6.6

# 1. Proporción de personas con estatura entre 160 y 170 cm
pnorm(170, mean = media, sd = desv) - pnorm(160, mean = media, sd = desv)
## [1] 0.4096521
# 2. Juan mide 0.5 desviaciones estándar más que la media
juan <- media + 0.5 * desv

# Proporción de personas más altas que Juan
1 - pnorm(juan, mean = media, sd = desv)
## [1] 0.3085375
# 3. Estatura en el percentil 90
qnorm(0.90, mean = media, sd = desv)
## [1] 179.0582
# 4. Probabilidad de que una persona mida más de 180 cm
prob_180 <- 1 - pnorm(180, mean = media, sd = desv)
prob_180
## [1] 0.07718815
# 5. Probabilidad de que exactamente 1 de 5 personas mida más de 180 cm
dbinom(1, size = 5, prob = prob_180)
## [1] 0.2798809

Problema 8

# Definir media y desviación estándar
media <- 24.2
desv <- 4.1

# 1. Probabilidad de Obesidad tipo 1 (IMC entre 30 y 34.9)
pnorm(34.9, mean = media, sd = desv) - pnorm(30, mean = media, sd = desv)
## [1] 0.07405756
# 2. Proporciones esperadas en cada rango de riesgo

# Bajo peso (<18.5)
bajo_peso <- pnorm(18.5, mean = media, sd = desv)

# Normal (18.5 a 24.9)
normal <- pnorm(24.9, mean = media, sd = desv) - pnorm(18.5, mean = media, sd = desv)

# Sobrepeso (25 a 29.9)
sobrepeso <- pnorm(29.9, mean = media, sd = desv) - pnorm(25, mean = media, sd = desv)

# Obesidad tipo 1 (30 a 34.9)
obesidad1 <- pnorm(34.9, mean = media, sd = desv) - pnorm(30, mean = media, sd = desv)

# Obesidad tipo 2 (35 a 39.9)
obesidad2 <- pnorm(39.9, mean = media, sd = desv) - pnorm(35, mean = media, sd = desv)

# Obesidad tipo 3 (>=40)
obesidad3 <- 1 - pnorm(40, mean = media, sd = desv)

# Mostrar resultados
bajo_peso
## [1] 0.08222741
normal
## [1] 0.4855552
sobrepeso
## [1] 0.3404213
obesidad1
## [1] 0.07405756
obesidad2
## [1] 0.004153194
obesidad3
## [1] 5.818293e-05