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Stat400Mid1_s2012_VersionA_sol

# Stat400Mid1_s2012_VersionA_sol - Statistics 400 Mathematics...

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Statistics 400 / Mathematics 463 Practice Exam 1 (BL1) Spring 2012 75 min Version A 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 60 different arrangements. T. 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

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2. [10 points] Suppose that E and F are two events such that P ( E ) = 0 . 3 , P ( F ) = 0 . 5 and P ( F | E ) = 1 / 3. (a) [5 points] Find P ( F 0 | E 0 ). P ( F | E ) = P ( E F ) /P ( E ) = 1 / 3, so P ( E F ) = 0 . 1, P ( E F ) = P ( E ) + P ( F ) - P ( E F ) = 0 . 7 and P ( E 0 F 0 ) = P (( E F ) 0 ) = 1 - 0 . 7 = 0 . 3. Therefore P ( F 0 | E 0 ) = P ( E 0 F 0 ) /P ( E 0 ) = 0 . 3 / 0 . 7 = 3 / 7.
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Stat400Mid1_s2012_VersionA_sol - Statistics 400 Mathematics...

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