1. Rerks –Ngarm et al describe interim results from a high profile vaccine trial carried out in Thailand that hit the main media resources worldwide. The trial had a good standard design, eligible participants were randomly split in to two arms, one given the putative vaccine, the other a placebo. The participants were followed for 3.5 years, at which point they were invited back to take an HIV test. The endpoint is HIV conversion. Let us just consider the vaccine arm. Of the 8197 eligible participants, 51 were infected during the trial period. We want to find the probability θ of infection over this time window for those vaccinated. Ignore the fact that the participants in this trail were not randomly selected.
a. If we were to use a non-Bayesian approach, say a frequentist approach, what would be the estimate for θ be?
b. Now develop a Bayesian model to find this estimate. Give details for your model and then use R to obtain the posterior distribution for θ. Use a flat prior of Be(1,1).
c. Now consider another non-uniform prior and derive the resulting posterior. Calculate the posterior point estimates, and plot the prior and posterior.

2. Download the data file “CommuteAtlanta” and attach it in R Studio. We want to construct the confidence interval for the mean commute time in Atlanta.
a. Draw a histogram for the time variable in the data set.
b. Find the point estimate (sample mean) from the original sample.
c. Create 1000 bootstrap samples and use them to find the 95% simple bootstrap confidence interval for commute time using the percentile based bootstrap CI (alternative CI on slides).
d. Also test the hypothesis that the average commute time is less than 30 minutes.

3. The number of particles emitted by a radioactive source during a ten second interval has the Poisson distribution with mean λ. The radioactive source is observed over twenty non- overlapping intervals of ten seconds each. The number of particles emitted during each interval are: 4, 1, 3, 1, 3, 3, 3, 1, 1, 1, 6, 4, 4, 2, 2, 1, 1, 4, 2, 5.
Find a 95% simple bootstrapping confidence interval for the mean number of particles emitted. Bootstrap 5000 times.
Now assume a Gamma prior for λ. Choose the parameters for Gamma such that the expected mean and variance for λ is 2. In R if we have a Gamma(a,b), then the mean is a/b and the variance is a/b2 . Since the data are from a Poisson distribution, using conjugate priors in a Bayesian framework we know that the Posterior is a Gamma distribution.
e. Derive the posterior Gamma distributions and identify the parameters.
f. Simulate the posterior distribution for λ in R.
g. Give the posterior mean from the simulation.
h. Plot both the prior and posterior densities by sampling from the prior and posterior Gammas along with the likelihood function. You can use the ‘density’ function.

1. Rerks –Ngarm et al describe interim results from a high profile...