---
title: "HW 3"
author: "Name here"
date: "Due February 1, 2018 at 5:00 PM"
output: html_document
---

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.