In a 2006 study published in The New England Journal of Medicine, 78 pairs of patients with Parkinson's disease were randomly assigned to receive treatment (which consisted of deep-brain stimulation of a region of the brain affected by the disease) or control (which consisted of taking a prescription drug). The researchers found that in 50 of 78 pairs, the patients who received deep-brain stimulation had improved more than their partner in the control group. The parameter of interest is , the probability of doing better on treatment than control.

Using a Bayesian perspective, using a Uniform[0, 1] prior distribution on .

(A) What is the posterior distribution of |y?

(b) Plot the posterior density f( | y).

(c) Calculate a 95% posterior credible interval for . If you use any program, make sure to provide your codes.

(d) What is the posterior probability that > 0.5?

(E) Based on your analysis in (e)-(h), do you agree that deep-brain stimulation is "more effective than medical management".