set.seed(123456)

curve(dchisq(x,1))

par(mfrow=c(2,2))

ybar2<-numeric(10000)
for(j in 1:10000) {
  sample<-rchisq(2,1)
  ybar2[j] <-mean(sample)
}
mean(ybar2)
var(ybar2)
plot(density(ybar2), xlim = c(0, 10), ylim = c(0.0, 1.6))
curve(dnorm(x,1,sqrt(1.0)), add=T, lty=2)

ybar10<-numeric(10000)
for(k in 1:10000) {
  sample<-rchisq(10,1)
  ybar10[k] <-mean(sample)
}
mean(ybar10)
var(ybar10)
plot(density(ybar10),xlim = c(0, 4), ylim = c(0.0, 1.6))
curve(dnorm(x,1,sqrt(0.2)), add=T, lty=2)

ybar20<-numeric(10000)
for(m in 1:10000) {
  sample<-rchisq(20,1)
  ybar20[m] <-mean(sample)
}
mean(ybar20)
var(ybar20)
plot(density(ybar20),xlim = c(0, 3), ylim = c(0.0, 1.6))
curve(dnorm(x,1,sqrt(0.1)), add=T, lty=2)

ybar30<-numeric(10000)
for(n in 1:10000) {
  sample<-rchisq(30,1)
  ybar30[n] <-mean(sample)
}
mean(ybar30)
var(ybar30)
plot(density(ybar30),xlim = c(0.3, 1.7), ylim = c(0.0, 1.6))
curve(dnorm(x,1,sqrt(0.0667)), add=T, lty=2)