Often in medical research, more than one group is doing work in the same area and report their results separately. It is of interest to be able to combine the results of multiple trials. Suppose that researchers at UZ carry out a trial of Treatment SX for psoriasis. They treat 20 patients and observe13 successes. Meanwhile over at UM the researchers have a different trial design for the sameTreatment SX. Patients are treated until they observe 5 successes. It turns out that it takes the UM group a total erollment of 20 patients to observe 5 successes. Researchers at UZ and UM are actually no longer on speaking terms after a recent collaboration in which they could not decide on authorship for a publication. So you may con?dently assume that the trials are carried out independently. Moreover no patient is treated more than once.

(a) Let X be the number of successes at UZ. What is the distribution of X?
(b) Let Y be the number of patients enrolled at UM. What is the distribution of Y ?
(c) Provide maximum likelihood estimates of individual success rates for the two trials?
(d) Assume that the success rate of Treatment SX is identical for the two trials. Call this success rate p:Provide a maximum likelihood estimate of p. In other words how can you use maximum likelihood to combine the information from the two trials to get a single estimate of p
(e) What is the asymptotic distribution of your estimate from (d)?