#Central Limit Theorem
#
#Histogram of averages of size 5, 25 from Mixture of Normals .5N(-3.5,1) + .5N(3.5,1) Population
#
par(mfrow=c(3,1))
nsamp=10000
n=5
xbar5=NULL
for(iter in 1:nsamp)
  {unif=runif(n)
   xnormLeft=rnorm(n,mean=-3.5)
   xnormRight=rnorm(n,mean=3.5)
   data=c(xnormLeft[unif<.5],xnormRight[unif>.5])
  xbar5=c(xbar5,mean(data))}
n=25
xbar25=NULL
for(iter in 1:nsamp)
  {unif=runif(n)
   xnormLeft=rnorm(n,mean=-3.5)
   xnormRight=rnorm(n,mean=3.5)
   data=c(xnormLeft[unif<.5],xnormRight[unif>.5])
  xbar25=c(xbar25,mean(data))}
plot(seq(-6,6,.001),.5*dnorm(seq(-6,6,.001),mean=-3.5)+.5*dnorm(seq(-6,6,.001),mean=3.5),type="l",
main="Mixture of Normals Population,  .5N(-3.5,1)+.5N(3.5,1)",cex.main=1.8,ylab="Density")
abline(v=0)
hist(xbar5,xlim=c(-4,4),main="Sample Mean, n=5",cex.main=1.8)
abline(v=0)
mtext("Mean=",at=c(-2,1000))
mtext(round(mean(xbar5),3),at=c(-1.3,1000))
mtext("SD=",at=c(2,1000))
mtext(round(sqrt(var(xbar5)),3),at=c(2.5,1000))
hist(xbar25,xlim=c(-4,4),main="Sample Mean, n=25",cex.main=1.8)
abline(v=0)
mtext("Mean=",at=c(-2,1000))
mtext(round(mean(xbar25),3),at=c(-1.3,1000))
mtext("SD=",at=c(2,1000))
mtext(round(sqrt(var(xbar25)),2),at=c(2.5,1000))
#