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2027Hmwk7 - ISyE 2027 R D Foley Probability with...

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ISyE 2027 Probability with Applications Fall 2011 R. D. Foley Homework 7 October 25, 2011 due Monday 1. A factory receives a particular part from 3 different suppliers A, B and C. The defect rate for parts from A is 1%, from B 2%, and from C 3%. Supplier A supplies 60% of the parts, and the remainder is split equally between B and C. (a) What fraction of the parts received at the factory are defective? (b) Given that a part is defective, what is the probability that it came from C? 2. Suppose that a particular medical test will come back positive 99% of the time if the person being tested is HIV positive. The test comes back negative 95% of the time if the person being tested is not HIV positive. (a) Suppose the prior 1 probability that person A is HIV positive is one in a million. Given that the test comes back positive, is the probability that A is HIV positive 99%? (b) If not, what is the probability that A is HIV positive given the test is positive? (c) Suppose the prior probability that person B is HIV positive is 1/10. Given that the test comes back positive, what is the probability that B is HIV positive? (d) Would it make sense to use this test to do mass screenings of every resident in the state of Georgia? 3. Suppose Y has a p.d.f. with support from (- 1, 1 ) . The p.d.f. is symmetric around zero. In the positive quadrant, the p.d.f. looks like a straight line from ( 0, h ) to ( 1, 0 ) where h is chosen appropriately. Compute (a) h , (b) Pr { - 1 / 2 < Y 6 1 / 2 } , (c) the mean of Y , (e) the variance of Y , (f) the mean of Y - E [ Y ] , (g) the mean of ( Y - E [ Y ]) 2 , and (h) E [( Y - 1 / 2 ) + ] where x + = max ( x , 0 ) . 4. Suppose X has c.d.f. F ( s ) = 1 π arctan ( s ) + 1 / 2 for - < s < Compute (a) Pr { - 1 < X 6 1 } , (b) the p.d.f. of X , (c) the median of X , and (d) the mean of X . 5. Suppose you go fishing in a small-landlocked lake, and you catch, tag and release 30 bluegill. A week later, you return to the lake and catch 25 bluegill, which you temporarily keep in a live well. Only 5 of the 25 had been tagged. (a) If there are n bluegill in the lake, what is the probability of 5 tagged bluegill among the 25 caught? (b) What assumptions did you make in deriving your answer to the previous question? (c) Can you find 1 That is, prior to receiving the results of the test.
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