A non-destructive quality control test on a military weapon system correctly detects a ?aw in the central electronic guidance subunit if one exists, or correctly accepts the system as fully functional if no ?aw exists, 85% of the time; it incorrectly identi?es a ?aw when one does not exist (a false positive), 5% of the time, and incorrectly fails to detect a ?aw when one exists (a false negative), 10% of the time. When the test is repeated 5 times under mostly identical conditions, if X1 is the number of times the test is correct, and X2 is the number of times it registers a false positive, the joint pdf of these two random variables is given as:
f (x1, x2) = 120 x1 !x2! 0.85x1 0.05x2 (5.97)
(i) Why is no consideration given in the expression in Eq (5.97) to the third random variable, X3 , the number of times the test registers a false negative?
(ii) From Eq (5.97), generate a 5 × 5 table of f (x1, x2) for all the possible outcomes
and from this obtain the marginal pdfs, f1 (x1) and f2 (x2). Are these two random variables independent?
(iii) Determine the expected number of correct test results regardless of the other results; also determine the expected value of false positives regardless of other results.
(iv) What is the expected number of the total number of correct results and false positives? Is this value the same as the sum of the expected values obtained in (iii)? Explain.