Question

Please answer in RStudio

2. Complete **Exercise 6.72** of the text. Assess your results by simulation as follows. Reset your random number seed to 4302019, then (a) simulate 10000 realizations of the minimum of two Uniform(0,1) random variables Cuse runif in R, (b) plot their histogram, (c) add the theoretical probability density function that you derived in 6.72(a) to your histogram, (d) compute the empirical mean and variance of the minimum order statistic from your simulation. Finally, compare the theoretical values of SE(U_1)s and SV(U 1)S that you computed in 6.72(b) to your simulation values from

Answer 1

REQUIRED CODE

set.seed(4302019)
#part a)
a<-runif(1000)
b<-runif(1000)
x<-c()
for(i in 1:1000)
x<-c(x,min(runif(2)))
#part B)
hist(x,breaks = 10)
#part c) HERE CORRECT DENSITY IS 2*(1-X)
d<-c(0:50)/50
e<-2*(1-d)
hist(x,breaks = 10,probability = TRUE)
lines(d,e)
#part d)
mean_min<-mean(x)
var_min<-var(x)
print(mean_min)
print(var_min)
#theoritical mean is 0.33
#theoritical variance is 0.056

______________________________________

R CODE OUTPUT

> source('~/.active-rstudio-document', echo=TRUE)

> set.seed(4302019)

> #part a)
> a<-runif(1000)

> b<-runif(1000)

> x<-c()

> for(i in 1:1000)
+ x<-c(x,min(runif(2)))

> #part B)
> hist(x,breaks = 10)

> #part c) HERE CORRECT DENSITY IS 2*(1-X)
> d<-c(0:50)/50

> e<-2*(1-d)

> hist(x,breaks = 10,probability = TRUE)

> lines(d,e)

> #part d)
> mean_min<-mean(x)

> var_min<-var(x)

> print(mean_min)
[1] 0.3405045

> print(var_min)
[1] 0.06019336

> #theoritical mean is 0.33
> #theoritical variance is 0.056