mid08sol - Midterm Exam 45-730 Probability Spring 2008 Name...

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Unformatted text preview: Midterm Exam 45-730 Probability Spring 2008 Name SOLUTION KEY This test is open notes, open book. Laptops are allowed but Internet communication is strictly forbidden. There are fourteen questions total: ten multiple-choice questions and four prob- lems. For each of the multiple-choice questions, select the one best answer in your judgment: marking more than one answer will result in no credit. For each of the remaining problems, please write your answer in fractional form (for example 3/4, 23/41, etc) or with at least four digits of accuracy (for example 0.7500, 0.5609, etc). Show your work. The test is one hour and fifty minutes long. Maximum score is 40 points. 1 PART I: Multiple choice questions (2 points each) 1. Let A 1 ,A 2 ,A 3 be three events. Which of the following statements is true? If A 1 ,A 2 ,A 3 are mutually exclusive then P ( A 1 A 2 A 3 ) = 1 If A 1 ,A 2 ,A 3 are mutually exclusive then P ( A 1 A 2 A 3 ) = 0 If A 1 ,A 2 ,A 3 are collectively exhaustive then P ( A 1 A 2 A 3 ) = 0 If A 1 ,A 2 ,A 3 are collectively exhaustive then P ( A 1 A 2 A 3 ) = 1 2. Airline customers arrive at the rerouting desk according to a Poisson process with arrival rate of 6 per 10 minute interval. The probability that an arriving person is seriously aggravated is 50%. What is the chance that more than 100 seriously aggravated customers arrived between 8 and 9 pm? POISSON(100, 36, 1) 1- POISSON(100, 36, 1) 1- POISSON(100, 60, 1) 1- POISSON(100, 18, 1) 3. In the same context as in the previous question, suppose that exactly 30 customers arrived in the first half hour of operations. What is the probability that at least half of them are seriously aggravated? 1- BINOMDIST(14, 30, 0.5, 1) 1- BINOMDIST(15, 30, 0.5, 1) 1- BINOMDIST(14, 15, 0.5, 1) POISSON(30, 18, 1) 4. Suppose X,Y are discrete random variables with E[ X ] = 1, var[ X ] = 12, E[ Y ] = 12, and var[ Y ] = 1. Which of the following statement is true? If X and Y are dependent, then E[ X + Y ] may be larger than 13 If X and Y are dependent, then var[ X + Y ] may be larger than 13 If X and Y are independent, then var[ X + Y ] may be larger than 13 If X and Y are independent, there is not enough information to compute var[ X + Y ] 2 5. The possible basic outcomes and corresponding probabilities of a random experiment are shown in the following table, the two entries ? indicating that these numbers are not known: outcome a b c d e probability...
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mid08sol - Midterm Exam 45-730 Probability Spring 2008 Name...

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