Problema 1


n <- 10
p <- 0.5

# -----------------------------------------------------------------------------# 
x  <- 0:n # Valores posibles
fx <- dbinom(x, size = n, prob = p) # Función de probabilidad (pmf)
Fx <- pbinom(x, size = n, prob = p) # Función de distribución (cdf)
# -----------------------------------------------------------------------------#

Probabilidades


dbinom(5, n, p)                     # P(X=5)                 
## [1] 0.2460938
pbinom(2, n, p)                     # P(X<=2)
## [1] 0.0546875
pbinom(4, n, p) - pbinom(3, n, p)   # P(3<=X<5)
## [1] 0.2050781
1-pbinom(8,n,p)                     # P(X=>8)
## [1] 0.01074219
barplot(fx, names.arg = x,
        xlab = "x", ylab = "f(x)",
        main = "Funcion de masa")

Problema 2


lambda <- 4

dpois(0,4)
## [1] 0.01831564
dpois(4,4)
## [1] 0.1953668
1-ppois(1,4)
## [1] 0.9084218
ppois(2,4)
## [1] 0.2381033

Gráfica


x   <- 0:15
fx  <- dpois(x, lambda) # Función de probabilidad (pmf)

barplot(fx, names.arg = x,
        xlab = "x", ylab = "f(x)",
        main = "Distribución Poisson (λ = 4)")

Problema 3


Binomial negativa —> Parámetros


N <- 100   # Poblacion 
K <- 20    # Evento: Exito
n <- 5     # Muestra 
p <- K / N # Probabilidad de éxito

Probabilidades


dnbinom(0, size = n, prob = p)      # P(X = 0)
## [1] 0.00032
dnbinom(6, size = n, prob = p)      # P(X = 6)
## [1] 0.01761608
1 - pnbinom(9, size = n, prob = p)  # P(X ≥ 10)
## [1] 0.8701604
pnbinom(12, size = n, prob = p)     # P(X ≤ 12)
## [1] 0.2417768
E <- n * (1 - p) / p
V <- n * (1 - p) / (p^2)

Funcion de masa acumulada—-> Binomial negativa


x  <- 1:20
fx  <- dnbinom(x, size = n, prob = p) 
fx
##  [1] 0.00128000 0.00307200 0.00573440 0.00917504 0.01321206 0.01761608
##  [7] 0.02214593 0.02657511 0.03070902 0.03439410 0.03752083 0.04002222
## [13] 0.04186940 0.04306567 0.04363988 0.04363988 0.04312647 0.04216810
## [19] 0.04083648 0.03920302
barplot(fx, names.arg = x,
        xlab = "x", ylab = "f(x)",
        main = "Distribucion binomial negativa ")

Problema 4


### Binomial —> Parámetros 

n <- 4
p <- 0.05

Probabilidades


dbinom(0, size = n, prob = p) # Ninguna con imperfecciones P(X=0)
## [1] 0.8145062
dbinom(1, size = n, prob = p) # Solo una con imperfecciones P(X=1) 
## [1] 0.171475

1 o más con imperfecciones P(X>=1)


1 - dbinom(0, size = n, prob = p) # 1 - P(X=0)
## [1] 0.1854938
1 - pbinom(0, size = n, prob = p) # 1 - P(X<1)
## [1] 0.1854938

Problema 5


lambda <- 8 # Ocho clientes por hora

1) 8:00 AM a 9:00 PM (1 hora) P(X ≤ 3)


dpois(5, lambda)  
## [1] 0.09160366

2) 2:30 PM a 3:30 PM (1 hora) P(X ≤ 3)


ppois(3, lambda = 8)
## [1] 0.04238011

3) 10:00 A.M a 12 A.M (2 Horas) P(X_2 = 2)


lambda2 <- 2*lambda
dpois(2, lambda) 
## [1] 0.0107348

4) Valor esperado entre 2 PM a 4:30 PM


lambda3 <- 2.5*lambda
lambda3
## [1] 20

Problama 6


n <- 105  # Cantidad de tiquetes vendidos    
p <- 0.90 # probabilidad de que la persona llegue al vuelo 

pbinom(100, # Cantidad de asientos en el avion 
       size = n, prob = p)
## [1] 0.9832837