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 annual snowfall totals can be obtained.

Below are the snowfall totals for Taos and Bridger Bowl for the last nine years.

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.