XI. RANDOMIZED DATA PROCESSING PART 1 (ONE DISEASE TESTER) 0.8 latesalarmprdaahllty 3 E 2 E 3 Ideal {1,0} pel'ionna'loe a 0 0.1 0.! 0.3 0.4 ll! 08 0.7 0.! 0.0 I detection probability Fig. l. A 2-d visualization of (paged :pFaiseAlm-m] for the original tester T. We have a simple disease tester T that tests whether or not a person has a disease. However, it gives poor perfmnance. We want to design a randomized data processor that improves the performance. The tester T outputs a (possibly incorrect) binary result X 6 {0,1}. If X = 1 it means the tester thinks the person has the disease. If X = 0 it means the tester thinks the person does not have the disease. Let H be the event that the person being tested has the disease. The disease tester has the following detection and false alarm probabilities (see Fig. ID: pdetect = P[X = lim = 0-2 a Plates = P[X = IIHC] = 0'85 We want to a design a function 3' that takes X as input and outputs a random variable Y 6 {0,1}. so Y = f(X). Dene the new detection and false alarm probability as: 113293.22 = PIY = 1|H] . 9?:3'39 = PIY = 1|H\"] a) Suppose Y = f(X) where f : {0,1} > {0,1} is a deterministic function. How many deterministic functions are there? Plot the (pggd, pe) operating points associated with each deterministic function. h) Assume we can randomize the choice of deterministic functions we use: If your part (a) has it functions f1(X), . . . , LAX), then we choose function f,- with some probability pi, where p,- 2 0 for all i E {1, ...,n.} and SL1 p,- = 1. The choice of which 3'. to use is made independently of whether or not the person has the disease. Let p33\" and 33??\" be the detection and false alarm probabilities under this randomized testing model. Design [n'obabilities that maximize pct subject to the constraint p'e 5 0.13. Plot your optimized operating point on the same graph as part (a). This is 013th Is this ethical