# Stat 6401
# Take-home Midterm
# Problem 3

#################################################################################
# CLT

# Normal

n = 1000		# sample size
k = 1000		# number of samples

mu = 5
sigma = 2
SEM = sigma/sqrt(n)

z_n = rnorm(n*k,mu,sigma)

x = matrix(z_n,n,k)		# This gives a matrix with the samples 
							# down the columns.

x.mean = ( apply(x,2,mean) - mu)/SEM

x.down = -4
x.up = 4

y.up = 0.5

hist(x.mean, prob= T, xlim= c(x.down,x.up), ylim= c(0,y.up),
      main= 'Sampling distribution of the sample mean, Normal case')

par(new= T)

x = seq(x.down,x.up,0.01)

y = dnorm(x,0,1)

plot(x, y, type= 'l', xlim= c(x.down,x.up), ylim= c(0,y.up), xlab="", ylab="")

#################################################################################
# Chi-Square

n = 50		# sample size
k = 1000		# number of samples

df = 1

mu = n*df
sigma = sqrt(2*df)*sqrt(n)

z_n = rchisq(n*k,df)

x = matrix(z_n,n,k)		# This gives a matrix with the samples 
							# down the columns.

x.sum = ( apply(x,2,sum) - mu)/sigma

x.down = -4
x.up = 4

y.up = 0.5

hist(x.sum, prob= T, xlim= c(x.down,x.up), ylim= c(0,y.up),
      main= 'Sampling distribution of the sample sum, Chi-Square case')

par(new= T)

x = seq(x.down,x.up,0.01)

y = dnorm(x,0,1)

plot(x, y, type= 'l', xlim= c(x.down,x.up), ylim= c(0,y.up), xlab="", ylab="")