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Unformatted text preview: Statistics 400 / Mathematics 463 Practice Exam 1 (BL1) Spring 2012 75 min Version B Solutions 1. True or False (No need to explain) [14 points] (a) For a continuous random variable, the density function should be continuous. F. For a continuous random variable, cdf should be continuous, not pdf. (b) Standard Cauchy distribution has a density f ( x ) = 1 π (1 + x 2 ) that is symmetric about the origin, so the mean and median both equal zero. F. Median is 0, but mean does not exist. (c) Consider two baseball players A and B. It is possible for batter A to have a higher average than batter B for each season during their careers and yet B could have a better overall average at the end of their careers. T. Simpson’s paradox. (d) Let X be of Uniform(a,b) distribution, then the value c that minimized E ( X c ) 2 is c = ( a + b ) / 2. T. Mean EX is the value to minimize the function E ( X c ) 2 . (e) If two events A and B are mutually exclusive, then they are also independent. F. Mutually exclusive implies that P ( A ∩ B ) = 0, while independence requires that P ( A ∩ B ) = P ( A ) P ( B ) which is not necessarily zero. (f) To scramble the word “ILLINI”, there are 50 different arrangements. F. Total number of arrangements is 6 3 , 2 , 1 = 60. (g) Let A and B be two events of positive probability. Then the conditional proba bility P ( A  B ) > P ( A ) if we know that P ( B  A ) > P ( B ). T. P ( B  A ) > P ( B ) implied that P ( A ∩ B ) > P ( A ) P ( B ), which in turn implies that P ( A  B ) > P ( A ). Actually P ( A  B ) > P ( A ) if and only if P ( B  A ) > P ( B ). 1 2. [10 points] Alex, Bob and Sam are on the same team when playing paintball. They all aim at the same target at this time. Suppose we know that Alex hits his target with probability 0.7, Bob misses 20% of the time and Sam has a chance of 0.4 to hit the target. Assume that their attempts are independent of each other....
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This note was uploaded on 03/14/2012 for the course STAT 400 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Kim
 Statistics, Probability

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