**Question 1** The Central Limit Theorem underpins a tremendous number of important results in statistical inference. It's also a somewhat bizarre and unexpected "truth" about how averages work.

For (almost any) probability distribution, find $\mu$ (the expected value) and $\sigma$ (the standard deviation). Once you have those two numbers, you can predict how far the average ($\bar{x}$) that you'll measure in a sample (of size $n$) will be from $\mu$ (typically about $\sigma/\sqrt{n}$). You can even put probabilistic bounds on how far $\bar{x}$ will be from $\mu$ (e.g. a 90% chance that $\bar{x}$ will be at most $1.65\sigma/\sqrt{n}$) since it sampling distribution will be approximately Normal *regardless of whatever PDF/PMF is making the data*.

The function `CLT.demo` in the chunk at the top of this document provides code that shows the sampling distribution of the sample average for samples of various sizes. It needs an argument named `PMF`, which is a vector of values giving the probability of observing a 0, a 1, a 2, etc.

a) In the R chunk below, I have defined `shape` to be a vector of 15 ones Replace these 1s with integers to define the shape of a custom PMF of your choosing. Try to make the shape weird! Once that is done, convert the shape into a PMF and left-arrow it into a vector named `PMF` and give a barplot of its distribution.

```{r Q1a}
shape <- c(1, 2, 1, 3, 1, 2, 1, 6, 1, 1, 2, 3, 4, 6, 5)
PMF <- shape/sum(shape)
barplot(PMF,names.arg=0:14)
```

b) Using `sample`, generate two random samples from this PMF (the first argument to `sample` will be `0:14`. One with a sample size of 25, another with a sample size of 5000. Provide histograms of each (adding the argument `breaks=-1:14)` to each call of histogram). Notice how in the smaller sample, the histogram only has a passing resemblence to the barplot of the PMF, but the histogram of the larger sample looks a lot like the PMF. **When we add more data to a sample, the histogram of its values look more and more like the probability distribution generating the data**

```{r Q1b}
sample1 <- sample(0:14,size=25,replace=TRUE,prob=PMF)
hist(sample1,breaks=-1:14)

sample2 <- sample(0:14,size=5000,replace=TRUE,prob=PMF)
hist(sample2,breaks=-1:14)

```

c) Run `CLT.demo(PMF)` (in the global environment once you read in the .RData file). Note that when we look at the sampling distribution of the possible sample averages that we might measure when we go collect data looks more and more like a Normal distribution as we consider samples of larger sizes. **When we consider samples with progressively larger sizes, the histogram of the possible averages we might measure from them looks more and more like a Normal curve**

```{r Q1c}
CLT.demo(PMF)

```

In this assignment, you have been provided with a text file of a se...