MATH 351 Homework 03

In this homework we will simulate $R$ sample means, each of which is determined from $N$ random variables, from the same distribution you worked with in Homework 02, let's call this distribution $X$. The Central Limit Theorem for the sample mean dictates that the shape of the (many) sample means will be approximately Normal, with expectation equal to $\mathbb{E}[X]$ and variance $\mathbb{V}[X] / N$. We will only need one for-loop, of length $R$, for this entire assignment.

1. For $R = 500$, set up a for-loop that does the following each iteration:
   1. generates $N$ (you choose the size) random variables from your distirbution (from Homework 02) $X$.
   2. stores the mean and variance of the random variables,
   3. approximates and stores the parameters you chose from Homework 02 for your distribution $X$ using only the mean and variance of the random variables
2. Make a histogram of the one sample of size $N$ of your random variables, and one histogram each for the $R$ means, variances, and estimated parameter(s).
3. On the histogram of the means, add a density plot of the expected Normal distribution. You'll need to calculate $\mathbb{E}[X]$ and $\mathbb{V}[X]$.