MATH 351 Homework 06

1. We'll make one plot, where each part adds a new piece of information onto the plot. First, pick a distribution and some number of random variables $N$.
   1. Using the theory of the Central Limit Theorem, make a density plot of the appopriate Normal distribution that approximates the distribution of the mean. Make a label for this curve. Notice, there is no random variables used here.
   2. Use the following pseudo-code to draw on your plot a small tick at the expectation of your distribution:
   plt.plot(E[X], [small_number], marker = '|', color ='c') where the pipe character is found just below delete/backspace and by holding shift.
   3. Generate 3 random samples of random variables, each of the same size $N$, from your distribution. From each, calculate the mean and the standard deviation. Draw approximating Normal distributions where you've used the estimated mean to replace the expectation and the estimated standard deviation to replace the standard deviation (square root of variance) of the distribution. Make all of these curves similar is some regard, same color and/or same linestyle. Make one label for all of these curves.
   4. Using the theory of the CLT, and without adding anything new to your plot, what happens to each of the three distributions which approximate the CLT as $N \to\infty$?
2. Applied statistics almost exclusively works with just one of the distributions which approximate the CLT. A common tool is the $95\%$ confidence interval, which is calculated with the following equation $\hat{\mu} \pm 1.96 \cdot \hat{\sigma} / \sqrt{N}$ where $\hat{\mu}$ and $\hat{\sigma}$ are data based estimates of the expectation and the standard deviation of the distribution, respectively. Perform a simulation experiment to see how often a $95\%$ confidence interval captures the true expectation. That is, generate $R$ confidence intervals and from each count if the true expectation falls within the confidence interval. Your goal is to esimate the probability $\mathbb{E}[1_A(\mathbb{E}[X])]$ where $A = (\hat{\mu} - 1.96 \cdot \hat{\sigma} / \sqrt{N}, \hat{\mu} + 1.96 \cdot \hat{\sigma} / \sqrt{N})$.
3. The expectation above, $\mathbb{E}[1_A(\mathbb{E}[X])]$, is actually quite nuanced. What exactly is random about the argument to the expectation?
4. Prove the result $\mathbb{V}[aX + b] = a^2\mathbb{V}[X]$ for a random variable $X$ and a constant $a$. Use the expectation definition of a variance and expand the squared term inside the expectation first, before substituting in $aX + b$. A middle point along this path should be the expression $\mathbb{V}[X] = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$.