thesamples <- rbinom(10000,1,0.5)
plot(thesamples)
themeans <- cumsum(thesamples)/seq_along(thesamples)
plot(themeans, type="l")
abline(h=0.5)

#################
#################
par(mfrow=c(2,2))
num <- 5
themeans <- rep(0,100)
for (iter in 1:1000){
  themeans[iter] <- mean(runif(num, 0, 1))
}
hist(themeans, xlim=c(0,1), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 10
themeans <- rep(0,100)
for (iter in 1:1000){
  themeans[iter] <- mean(runif(num, 0, 1))
}
hist(themeans, xlim=c(0,1), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 20
themeans <- rep(0,100)
for (iter in 1:1000){
  themeans[iter] <- mean(runif(num, 0, 1))
}
hist(themeans, xlim=c(0,1), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 40
themeans <- rep(0,100)
for (iter in 1:1000){
  themeans[iter] <- mean(runif(num, 0, 1))
}
hist(themeans, xlim=c(0,1), col="blue", breaks=100, main=paste("number in sample = ", num))

################################
### num gives the number of samples
num <- 5
themeans <- rep(0,100)
for (iter in 1:1000){
  themeans[iter] <- mean(rexp(num, 1))
}
hist(themeans, xlim=c(0,4), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 10
themeans <- rep(0,100)
for (iter in 1:1000){
  themeans[iter] <- mean(rexp(num, 1))
}
hist(themeans, xlim=c(0,4), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 20
themeans <- rep(0,100)
for (iter in 1:1000){
  themeans[iter] <- mean(rexp(num, 1))
}
hist(themeans, xlim=c(0,4), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 40
themeans <- rep(0,100)
for (iter in 1:1000){
  themeans[iter] <- mean(rexp(num, 1))
}
hist(themeans, xlim=c(0,4), col="blue", breaks=100, main=paste("number in sample = ", num))

#################################
#################################
#################################

### num gives the number of samples
par(mfrow=c(2,2))
num <- 5
thesums <- rep(0,100)
for (iter in 1:1000){
  thesums[iter] <- sum(runif(num, 0, 1))
}
thesums.scaled <- (thesums - 0.5*num)/sqrt(num/12)
hist(thesums.scaled, xlim=c(-3,3), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 10
thesums <- rep(0,100)
for (iter in 1:1000){
  thesums[iter] <- sum(runif(num, 0, 1))
}
thesums.scaled <- (thesums - 0.5*num)/sqrt(num/12)
hist(thesums.scaled, xlim=c(-3,3), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 20
thesums <- rep(0,100)
for (iter in 1:1000){
  thesums[iter] <- sum(runif(num, 0, 1))
}
thesums.scaled <- (thesums - 0.5*num)/sqrt(num/12)
hist(thesums.scaled, xlim=c(-3,3), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 40
thesums <- rep(0,100)
for (iter in 1:1000){
  thesums[iter] <- sum(runif(num, 0, 1))
}
thesums.scaled <- (thesums - 0.5*num)/sqrt(num/12)
hist(thesums.scaled, xlim=c(-3,3), col="blue", breaks=100, main=paste("number in sample = ", num))

################################
### num gives the number of samples
num <- 5
thesums <- rep(0,100)
for (iter in 1:1000){
  thesums[iter] <- sum(rexp(num, 1))
}
thesums.scaled <- (thesums - num)/sqrt(num)
hist(thesums.scaled, xlim=c(-3,3), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 10
thesums <- rep(0,100)
for (iter in 1:1000){
  thesums[iter] <- sum(rexp(num, 1))
}
thesums.scaled <- (thesums - num)/sqrt(num)
hist(thesums.scaled, xlim=c(-3,3), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 20
thesums <- rep(0,100)
for (iter in 1:1000){
  thesums[iter] <- sum(rexp(num, 1))
}
thesums.scaled <- (thesums - num)/sqrt(num)
hist(thesums.scaled, xlim=c(-3,3), col="blue", breaks=100, main=paste("number in sample = ", num))

num <- 40
thesums <- rep(0,100)
for (iter in 1:1000){
  thesums[iter] <- sum(rexp(num, 1))
}
thesums.scaled <- (thesums - num)/sqrt(num)
hist(thesums.scaled, xlim=c(-3,3), col="blue", breaks=100, main=paste("number in sample = ", num))