4. In this problem, we will attempt to evaluate the information that a cancer test cited in Murphy's textbook Example 2.

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4 In This Problem We Will Attempt To Evaluate The Information That A Cancer Test Cited In Murphy S Textbook Example 2 1 (75.62 KiB) Viewed 31 times

4. In this problem, we will attempt to evaluate the information that a cancer test cited in Murphy's textbook Example 2.2.3.1 provides to a patient. Let X be a Bernoulli random variable indicating whether or not a patient has cancer. Similarly, let Y be a Bernoulli random variable indicating whether the test is positive or not. Based on clinical studies, the test is found to have a detection probability of about 0.8 (i.e, the probability that the test for somebody with cancer turns out to be positive). At the same time, the test has a false positive probability of 0.1 (i.e., the probability that the test for somebody without cancer turns out to be positive). The prior probability that a patient has cancer is found to be 0.004. I(X;Y). (a) Evaluate the quantity r = *(X); the normalized information conveyed by Y about X. (b) What is the maximum false positive probability (for the same probability of detection of 0.8) that can be tolerated if we wish to achieve r=0.7? Calculate down to 6 decimal places. You will need to use a computer for this. (c) For any general X and Y, show that r < 1. (d) For any general X and Y , when is r = 1? (e) For any general X and Y , when is r = 0?