9. [10 marks] One in 5000 individuals in a certain population has the human immunodeficiency virus...

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9. [10 marks] One in 5000 individuals in a certain population has the human immunodeficiency virus (HIV). A test for the virus is devised and the result of the test is either + or - suggesting respectively the presence or absence of the virus. Clinical trials confirm the test to be 99% correct so that P[+ | H] = .99 = P[-|H] where H and HC represent respectively presence and absence of the virus. A) What is the probability that a person with HIV gets a negative response to the test. B) What is the probability that a person who gets a + on the test, has HIV. C) The 'false positive' situation is one in which a person who gets a + in the test is in fact HIV free. Calculate this probability and explain why it is so high. 8. [15 marks] A fair six-sided die is rolled and a fair coin is tossed. A head counts for 1 and a tail for 0. The die face and the 'value' of the coin toss are added to form the random variable X. (eg. rolling a 3 and tossing a head gives X = 4.) (a) Write down the sample space Sx for X. (b) Calculate the PMF for X. (A table might help here.) (c) Calculate the value of the CDF for X at X = 4.5 (d) Find the expected value of X, E(X). (e) Calculate the probability that X is even given that it is less than seven. 9. [10 marks] One in 5000 individuals in a certain population has the human immunodeficiency virus (HIV). A test for the virus is devised and the result of the test is either + or - suggesting respectively the presence or absence of the virus. Clinical trials confirm the test to be 99% correct so that P[+ | H] = .99 = P[-|H] where H and HC represent respectively presence and absence of the virus. A) What is the probability that a person with HIV gets a negative response to the test. B) What is the probability that a person who gets a + on the test, has HIV. C) The 'false positive' situation is one in which a person who gets a + in the test is in fact HIV free. Calculate this probability and explain why it is so high. 8. [15 marks] A fair six-sided die is rolled and a fair coin is tossed. A head counts for 1 and a tail for 0. The die face and the 'value' of the coin toss are added to form the random variable X. (eg. rolling a 3 and tossing a head gives X = 4.) (a) Write down the sample space Sx for X. (b) Calculate the PMF for X. (A table might help here.) (c) Calculate the value of the CDF for X at X = 4.5 (d) Find the expected value of X, E(X). (e) Calculate the probability that X is even given that it is less than seven.