#5. Central Limit Theorem
#
#Histogram of averages of size 5, 25 from Uniform (0,1) Population (Mean = 1/2, Variance = 1/12).
#Overlay the appropriate normal density.
#
par(mfrow=c(3,1))
nsamp=10000
n=5
xbar5=NULL
for(iter in 1:nsamp)
  xbar5=c(xbar5,mean(runif(n)))
n=25
xbar25=NULL
for(iter in 1:nsamp)
  xbar25=c(xbar25,mean(runif(n)))
plot(seq(0,1,.001),dunif(seq(0,1,.001)),type="l",
ylim=c(0,1),main="Uniform(0,1) Population (Mean = 1/2, Variance = 1/12)",cex.main=1.8,ylab="Density")
abline(v=0)
abline(v=1)
abline(v=.5)
hist(xbar5,freq=F,xlim=c(0,1),ylim=c(0,3.5),main="Sample Mean, n=5",cex.main=1.8)
abline(v=.5)
mtext("Mean =",at=c(.1,1000))
mtext(round(mean(xbar5),3),at=c(.2,1000))
mtext("SD =",at=c(.8,1000))
mtext(round(sqrt(var(xbar5)),3),at=c(.9,1000))
x=seq(0,1,.001)
curve(dnorm(x,mean=.5,sd=1/sqrt(5*12)),add=T)
hist(xbar25,freq=F,xlim=c(0,1),ylim=c(0,7),main="Sample Mean, n=25",cex.main=1.8)
abline(v=.5)
mtext("Mean =",at=c(.1,1000))
mtext(round(mean(xbar25),3),at=c(.2,1000))
mtext("SD =",at=c(.8,1000))
mtext(round(sqrt(var(xbar25)),3),at=c(.9,1000))
curve(dnorm(x,mean=.5,sd=1/sqrt(25*12)),add=T)
#