(a) X, ...Xn is a random sample of the time-to-infection in the experiment arm (Arm 1). Assuming the drug. efficacy to prevent a relapse is 100%, these reoccurrences of infections are all new musquito bites. If we assume the time to a new mosquito bite follows an exponential distribution with pat fx (x) = B exp (-Bx), 0O Show that the MLE of B is B = X"), where X=n'' Ê Xi and that √ñ( ß-B)→INIU, Baj (b) say we want to know if the new infection rate is higher than 0.1 per day. Testing Ho: B = 0.1 vs Hi: B>0.1, derive the uniformly most powerful level & test for the hypothesis. Find the cutoff in the rejection region explicitly. (c) suppose Y.... Yn is a random sample of time in days of an infection in the control arm (Arm 2). The reoccurences of the infections may be dire to relapse or a new mosquito bite. If we assume the time to infection follows an exponential distribution with pdf: L∞ r>o fy(4) = (B+r) exp |-|B+rdy) okyaw осуко i = 1 where r is the relapse rate. Snow that Ÿ''-x" is a consistent estimator of r, where Y=n'Ê. Yi Also show that it is asymptotically normal. That is show that √n (Y"- X" - r) → N 10,0%) and express o explicitly. -1 d

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