Please use D2L to turn in both the HTML output and your R Markdown file in.

### Q1. (8 pts)

The probability density function of the beta distribution can be written as: \[p(x|\alpha, \beta) = x^{\alpha -1} (1-x)^{\beta -1} \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)},\] for \(x \in [0,1]\).

Plot the density function for a beta distribution with:

- \(\alpha = 1,\) \(\beta = 1\)
- \(\alpha = 15,\) \(\beta = 3\)
- \(\alpha = 150,\) \(\beta = 30\).

How do the values of \(\alpha\) and \(\beta\) change the distribution?

### Q2. (8 pts)

Simulate 1000 observations from each of the distributions below:

- Normal(7,9)
- Uniform(0,1)
- Beta(1,1)
- Beta(15,3)

Use the samples from each distribution to calculate the mean and variance. How do your calculations compare with the known mean and variance from these distributions?

### Q3. (4 pts)

A main emphasis in Bayesian statistics is to think about the entire distribution of possibilities, rather than a point interval.

- Discuss how inferences could be different for two distributions with the same mean, but different variances.
- Discuss how inferences could be different for two distributions with the same mean and variance, but different HDI.