Question

To better understand how the Law of Large Number (LLN) and the Central Limit Theory (CLT) works, we can use the following simulation to demonstrate them. (Note: attach the computer codes which implement the analysis in your answer) To demonstrate the Lindberg-Levy CLT, please generate 1000 sets of independent random samples, each has 1000 observations, from i.i.d X^2(5) distribution. First choose one set of the random samples and plot a histogram. Do you see a X^2(5) distribution? Denote the sample mean of each set of random samples as Z, and plot the histogram of Z. Do you see a normal distribution? What are the empirical mean and empirical variance of Z? Are they close to what CLT predicts?

Answer 1

1000 samples of 1000 observations of chisquare RVs are generated and mean of the observations is plotted as below.

Below the R code for one set of the random sample.

m <- 1000
n <- 1000
df <- 25
TC <- array (dim = c(m,n))
MTC <- array (dim = m)
for (i in 1:m)
{
TC[i,] <- rchisq(n,df)
MTC[i] <- mean(TC[i,])
}

hist(TC[4,],prob = T, breaks = 50, xlab = "X", main = "Chisquare Distribution")
curve(dchisq(x,df), col="darkred", lwd=2, add=TRUE)

We see a Chisquare distribution.

Histogram of the mean is plotted below. Code below.

m <- 1000
n <- 1000
df <- 25
TC <- array (dim = c(m,n))
Z <- array (dim = m)
for (i in 1:m)
{
TC[i,] <- rchisq(n,df)
Z[i] <- mean(TC[i,])
}

hist(Z,prob = T, breaks = 50, xlab = "Z", main = "Theoretical and Empirical Distribution")
curve(dnorm(x,df, sqrt(2*df/n)), col="darkred", lwd=2, add=TRUE)

We see a normal distribution.

The empirical mean is 24.99798 and variance  0.0527