7.1.7i A test for diabetes is a measurement X of a person's blood sugar level following an overnight fast. For a healthy person, a blood sugar level X in the range of 70- 110 rng/dl is considered normal. When a measurement X is used as a test for diabetes, the result is called positive (event T- if X > 140; the test is negative (event T) if X < 110, and the test is ambiguous (event T°) if < 110 < 140.
Given that a person is healthy (event H), blood sugar measurement X is the Gaussian (90, 20) random variable. Given that person has diabetes, (event D), X is the Gaussian (60,40) random variable. A rand only chosen person is healthy with probability P[H] = 0.9 or has diabetes with probability P[D] = 0.1.
(a) What is the conditional PDF fx|H,(x)?
(b) Calculate the conditional probabilities P[T+[H], and P[T-|HJ.
(c) Find P[H|T-, the conditional probability that a person is healthy given the event of a negative test.
(d) When a person has an ambiguous test result (7), the test is repeated, possibly many tunes, until either a J) positive T+ or negative T result is obtained. Let N denote the number of times the test is given. Assuming that for a given person the result of each test is given pendent of the result of all other tests, find the conditional PMF of N given event H that a person is healthy. Note that N = I if the person has a positive T+ or negative T- result on the first test.