1. Probability (20 pts, page 1) Please include all your solutions for this question in page 1 of your PDF. (a) Show that if two variables a and y are independent, then their covariance is zero. (b) Table 1 represents a hypothetical study of a test for HIV in a population of intravenous drug users, some carrying HIV and others not. "+" denotes a positive outcome from the test and "-" denotes a negative outcome. Outcome with HIV without HIV Total + 72 12 84 3 71 74 Total 75 83 158 Table 1.1. HIV test summary for intravenous drug users i. Calculate the probability of test being positive given the patient has HIV. ii. Calculate the probability of test being negative given the patient has HIV. iii. If the prevalence of HIV is 12% in intravenous drug users (12 cases per 100 patients), what is the probability of the patient having HIV after a positive test? A negative test