---
title: "HW 6"
author: "Name here"
date: "Due Friday October 26, 2018"
output: html_document
---

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


### Q1.
This question is revisiting Q2 from HW4 but fitting these values as a multivariate response. After enjoying numerous Bridger Bowl powder days during your time at MSU, you have received a job offer in New Mexico. You decision hinges on the quality of snow at your potential new home mountain, Taos.

According to [https://www.onthesnow.com/new-mexico/taos-ski-valley/historical-snowfall.html](https://www.onthesnow.com/new-mexico/taos-ski-valley/historical-snowfall.html) annual snowfall totals can be obtained.

Below are the snowfall totals for Taos and Bridger Bowl for the last nine years.
```{r}
taos <- c(78,192,169,179,191,204,197,116,195)
bridger <- c(271,209,228,166,316,254,344,319,247) 
```

Specifically you are interested in computing three probabilistic statements.

1. $Pr[\theta_{bridger} > \theta_{taos}]$ where $\theta_{bridger}$ is the mean annual snow fall at Bridger Bowl and $\theta_{taos}$ is the mean annual snow fall at Taos.
2. $Pr[\theta_{bridger} > 250]$ 
3. $Pr[\theta_{taos} > 200]$

#### a. (10 pts)
Write out the sampling model and define all parameters. Make sure to specify values in the prior distributions.

#### b. (30 pts)
Fit a multivariate Gibbs sampler. Include your code in-line here and plot: the posterior distribution for $\theta$ (this will be two dimensional) and the marginal distributions for $\theta_{bridger}$ and $\theta_{taos}$. Also include the posteriors means for the components in the covariance matrix.

### c. (10 pts)
Answer the three probabilistic questions above.