rate <- 1/877803
size <- 40
average <- NULL
for(i in 1:1000)
    average <- c(average, mean(rexp(n = size, rate = rate)))
library(ggplot2)
qplot(rexp(n = 1000, rate = rate), geom = "density")

theo_mean <- 1/rate
cat("Media teorica de los ingresos en la población ", theo_mean)
## Media teorica de los ingresos en la población  877803
sample_mean <- mean(average)
cat("Media muestral de los mil datos simulados", sample_mean)
## Media muestral de los mil datos simulados 880381.3
thvar <- (rate * sqrt(size)) ^ -2
cat("Varianza teorica de los ingresos en la población ", thvar)
## Varianza teorica de los ingresos en la población  19263452670
samvar <- var(average)
cat("Varianza muestral de los mil datos simulados", samvar)
## Varianza muestral de los mil datos simulados 19119476205
library(ggplot2)
dfRowMeans<-data.frame(average)
mp<-ggplot(dfRowMeans,aes(x=average))
mp<-mp+geom_histogram(fill="gray",color="black",aes(y = ..density..))
mp<-mp + labs(title="Densidad para los 40 números de la distribución exponencial", x="Promedios para los 40 números", y="Densidad")
mp<-mp + geom_vline(xintercept=sample_mean,size=1.0, color="black")
mp<-mp + stat_function(fun=dnorm,args=list(mean=sample_mean, sd=sqrt(samvar)),color = "blue", size = 1.0)
mp<-mp + geom_vline(xintercept=theo_mean,size=1.0,color="yellow",linetype = "longdash")
mp<-mp + stat_function(fun=dnorm,args=list(mean=theo_mean, sd=sqrt(thvar)),color = "red", size = 1.0)
mp
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

##Ejercicios

1/sqrt(2*pi)
## [1] 0.3989423
a <- 0
library(ggfortify)
ggdistribution(dnorm, seq(0-3*1, 0+3*1, 0.1), mean = 0, sd = 1, colour = "blue", p = ggdistribution(dnorm, seq(a-0.05, a+0.05, 0.1), mean = 0, sd = 1, colour = "blue", fill = "blue"))

dnorm(x = a, mean = 0, sd = 1)
## [1] 0.3989423
densidad.normal.aproximada <- function(epsilon) pnorm(q = a + epsilon, mean = 0, sd = 1, lower.tail = TRUE) - pnorm(q = a - epsilon, mean = 0, sd = 1, lower.tail = TRUE)
densidad.normal.aproximada(0.5)
## [1] 0.3829249

mean = 2.44, sd = 2.84 y a = 1.15

##Ejercicio

a <- 1.15
library(ggfortify)
ggdistribution(dnorm, seq(0-3*1, 0+3*1, 0.1), mean = 2.44, sd = 2.84, colour = "blue", p = ggdistribution(dnorm, seq(0-3*1, a, 0.1), mean = 2.44, sd = 2.84, colour = "blue", fill = "blue"))

pnorm(q = 1.15, mean = 2.44, sd = 2.84, lower.tail = TRUE)
## [1] 0.3248333
library(ggfortify)
ggdistribution(pnorm, seq(0-3*1, 0+3*1, 0.1), mean = 2.44, sd = 2.84, colour = "blue", p = ggdistribution(pnorm, seq(a-0.05, a+0.05, 0.1), mean = 2.44, sd = 2.84, colour = "blue", fill = "blue"))

##Probabilidad entre dos valores

a <- 1.15
b <- 2
library(ggfortify)
ggdistribution(dnorm, seq(0-3*1, 0+3*1, 0.1), mean = 0, sd = 1, colour = "blue", p = ggdistribution(dnorm, seq(a, b, 0.1), mean = 0, sd = 1, colour = "blue", fill = "blue"))

pnorm(q = b, mean = 0, sd = 1, lower.tail = TRUE) - 0.5
## [1] 0.4772499
pnorm(q = b, mean = 0, sd = 1, lower.tail = TRUE) - pnorm(q = a, mean = 0, sd = 1, lower.tail = TRUE)
## [1] 0.1023218
library(ggfortify)
ggdistribution(pnorm, seq(0-3*1, 0+3*1, 0.1), mean = 0, sd = 1, colour = "blue", p = ggdistribution(pnorm, seq(b-0.05, b+0.05, 0.1), mean = 0, sd = 1, colour = "blue", fill = "blue", p = ggdistribution(pnorm, seq(a-0.05, a+0.05, 0.1), mean = 0, sd = 1, colour = "blue", fill = "blue")))

Las barras son las probabilidades acumuladas, una es la de cero que es del 50% y la otra de 2 del 100%. Se le resta el acumulado para determinar la probabilidad de cero a dos