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Unformatted text preview: Statistics 400 / Mathematics 463 Practice Exam 1 (BL1) Spring 2012 Solutions 1. True or False (No need to explain) [8 points] (a) For a continuous random variable, the cumulative distribution function should be continuous. T (b) The quantity median is the same as mean. F (c) Two events A and B are independent if the conditional probability P ( A | B ) is same as P ( A ), provided that P ( B ) > 0. T (d) Event A and its complement A are mutually exclusive and mutually exhaustive. T 2. [10 points] Suppose that E and F are two events such that P ( E ) = 0 . 7 and P ( F ) = . 5. (a) [5 points] Prove that P ( E ∩ F ) ≥ . 2. 1 ≥ P ( E ∪ F ) = P ( E )+ P ( F )- P ( E ∩ F ) . Therefore P ( E ∩ F ) ≥ . 7+0 . 5- 1 = 0 . 2. (b) [5 points] Suppose we know P ( E ∩ F ) = 0 . 3, find out P ( E | F ). P ( E | F ) = P ( E ∩ F ) P ( F ) = P ( F )- P ( E ∩ F ) P ( F ) = . 5- . 3 . 5 = 2 / 5 1 3. [10 points] Every day, Tom makes a return trip to campus either by bus, or by bike. If it is raining, he takes the bus with probability 0.9 and bikes with probability 0.1. If it is sunny, he takes the bus with probability 0.4, and bikes with probability 0.6. Suppose it rains with probability 0.15 and is sunny with probability 0.85....
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