### Central Limit Theorem
# Take samples from a normal population. Plot a histogram of the sampling
# distribution of x.bar.

m <- 50
sd <- 10

par(mfrow=c(2,2))

x <- (0:100)

p.norm <- dnorm(x,m,sd)
plot(x,p.norm,main="normal density",type="l",xlim=c(0,100))

x <- matrix(rnorm(3000,m,sd),nrow=30)
x.bar <- apply(x,2,mean)

hist(x.bar,main="sampling distribution of x.bar",xlim=c(40,60))

plot(density(x.bar,width=5),main="sampling distribution of x.bar",type="l",xlim=c(0,100))

# Take samples from a beta population. Plot a histogram of the sampling
# distribution of x.bar.

beta1 <- 5
beta2 <- 1


par(mfrow=c(2,2))

x <- (1:99)/100

p.beta <- dbeta(x,beta1,beta2)
plot(x,p.beta,main="beta density",type="l")

x <- matrix(rbeta(3000,50,10),nrow=30)
x.bar <- apply(x,2,mean)

hist(x.bar,main="sampling distribution of x.bar",xlim=c(0.6,1))

plot(density(x.bar,width=0.2),main="sampling distribution of x.bar",type="l",xlim=c(0,1))