A test for diabetes is a measurement of a person’s blood sugar level following an overnight fast. For a healthy person, a blood sugar level in the range of 70 − 110 m/ is considered normal. When a measurement is used as a test for diabetes, the result is called positive (event T +) if ≥ 140; the test is negative (event T −) if ≤ 110, and the test is ambiguous (event T 0 ) if 110 < < 140. Given that a person is healthy (event H), a blood sugar measurement is the Gaussian ( = 90, = 20) random variable. Given that a person has diabetes, (event ), is the Gaussian(160, 40) random variable. A randomly chosen person is healthy with probability P[H] = 0.9 or has diabetes with probability P[] = 0.1.
a. What is the conditional PDF |H(x)?
b. In terms of the Φ( . ) function, find the conditional probabilities P[T+|H], and P[T−|H].
c. Find the conditional probability P[H|T−] that a person is healthy given the event of a negative test.
d. When a person has an ambiguous test result, (T 0 ) the test is repeated, possibly many times, until either a 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 independent of the result of all other tests, find the conditional PMF of N given event H that a person is healthy. Note that N = 1 if the person has a positive T+ or negative result T− on the first test.