stat610_test1_2006

stat610_test1_2006 - STATGlO Semester I 2006—2007 Test...

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Unformatted text preview: STATGlO, Semester I 2006—2007 Test “1 (Date: 10/6/2006) 1. Let A, B, and C be three arbitrary events. Find the probability that exactly one of these three events will occur. Pimemb piAéuec u as) ;: pm) ’tPiis)+PCL) ~2g>(’Ai3>12.p(Aci.—;p( BC) er sprang) 2. A box has three coins. One has two heads, one has two tails, and the other is a fair coin with one head and one tail. A coin is chosen at random, is flipped, and comes up heads. What is the probability that if it is thrown another times it will come up heads? 0, p, MM pm) A": p( iii-[42.58) ~' (6) + (as. 11) Pic) 3. A new test has been devised for detecting a particular type of cancer. If the test is applied to a person who has this type of cancer, the probability that the person will have a positive reaction is 0.95 and the probability that the person will have a negative reaction is 0.05. If the test is applied to a person who does not have this type of cancer, the probability that the person will have a positive reaction is 0.05 and the probability that the person will have a negative reaction is 0.95. Suppose that in the general population, one person out of every 10000 people has this type of cancer. If a person selected at random has a positive reaction to the test, what is the probability that the person has this type of cancer? pCc)P(+ic~,) Pm \>(+[c) .3. 95¢ > poi-Ia“) H («3128 ‘ :2 C 2 . 3 ' fl— K 0-6.? Its—exa‘Ij-‘l— (OW 4. Let Z be the rate at which customers are served in a queue. Assume that Z has the pdf f(2) = 26—22 for z > 0, 0 otherwise. Find the probability density function (pdf) of T 2 1 /Z . (hag? .1 7352”?- A 2 .~2- .L 2. .2: “5:” 5. Suppose that X is a random Variable for which the moment generating function (mgf) is as follows: 1 W) = 6(4+ at + 64) for ——00 < t < oo. 0 (a) Find the probability distribution of X. (b) Find the mean and variance of X. ...
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