A researcher develops a new version of a treatment and wants to know the value of the parameter 0 describing the probability that it will cure a given patient. To test the new treatment, three trial studies were conducted and the results are as follows; study 1 2 3 15 20 16 number of patients number of cured patients 13 17 12 A Beta prior, Beta(a = 4, = 6), is assigned for 0. i.e., 0~ 4, B = 6). (a) Find the posterior density for 0 and write it in ""notation. Beta(a [3 marks] (b) Given the results from the three studies, find the expected value for the cure probability and its 90% credible interval. The expected value should be calculated with a formula, and the credible interval should be calculated in R. [2 marks] (c) A standard medical treatment that has been used for many years is known to cure a disease in 70% of patients. Is the new treatment better, and if so, by how much? [2 marks] (d) The new treatment is given to 13 new patients. Find the probability that at least 11 patients are cured given the study result. [4 marks]