---
title: "HW 5"
author: "Name here"
date: "Due Friday October 5, 2018"
output: html_document
---

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

### Q1. (5 pts) 
Sketch out the steps for a Gibbs sampler algorithm. 

### Q2. (45 pts)
Simulating data is a key step in verifying your algorithms are working correctly. This will be more apparent as we start studying sophisticated hierarchical models.

#### a. (5 pts) 
Simulate 100 observations from a standard normal distribution and plot a histogram of your data.

#### b. (5 pts)
Select and state prior distributions for $\theta$ the mean of the normal distribution and $\sigma^2$ the variance (or alternatively you may parameterize your model using the precision term).

#### c. (10 pts)
Implement a Gibbs sampler to simulate from the joint posterior distribution $p(\theta,\sigma^2|y_1, \dots, y_{100})$. Create a plot of the joint posterior distribution.

#### d. (10 pts)
 Plot trace plots and histograms of the marginal posterior distributions for $\theta$ and $\sigma^2$. Include the true values on these figures. Comment on the figures.

#### e. (10 pts)
 Use your MCMC samples to create a posterior predictive distribution. Compare the data and your posterior predictive distribution using a QQ plot `qqnorm()`. Comment on the figure.