set.seed(123456)

curve(dunif(x,min=0, max=1))

par(mfrow=c(2,2))

ybar2<-numeric(10000)
for(j in 1:10000) {
  sample<-runif(2,min=0, max=1)
  ybar2[j] <-mean(sample)
}
mean(ybar2)
var(ybar2)
plot(density(ybar2), xlim = c(0, 1), ylim = c(0,8))
curve(dnorm(x,0.5,sqrt(0.0417)), add=T, lty=2)

ybar10<-numeric(10000)
for(k in 1:10000) {
  sample<-runif(10,min=0, max=1)
  ybar10[k] <-mean(sample)
}
mean(ybar10)
var(ybar10)
plot(density(ybar10),xlim = c(0, 1), ylim = c(0,8))
curve(dnorm(x,0.5,sqrt(0.0083)), add=T, lty=2)

ybar20<-numeric(10000)
for(m in 1:10000) {
  sample<-runif(20,min=0, max=1)
  ybar20[m] <-mean(sample)
}
mean(ybar20)
var(ybar20)
plot(density(ybar20),xlim = c(0, 1), ylim = c(0, 8))
curve(dnorm(x,0.5,sqrt(0.0042)), add=T, lty=2)

ybar30<-numeric(10000)
for(n in 1:10000) {
  sample<-runif(30,min=0, max=1)
  ybar30[n] <-mean(sample)
}
mean(ybar30)
var(ybar30)
plot(density(ybar30),xlim = c(0, 1), ylim = c(0, 8))
curve(dnorm(x,0.5,sqrt(0.0028)), add=T, lty=2)