Practice Midterm11

Practice Midterm11 - Midterm 1 Solution Stat416/Math416...

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Midterm 1 Solution Stat416/Math416, Spring 2005 J. Li The number of points assigned to each question is indicated before the question. The full score for the exam is 100. 1. (20) There are two coins. One is a fair coin and the other is a biased coin with probability of obtaining head P ( H )=0 . 8. A coin is randomly chosen from the two and ﬂipped three times. What is the probability that three heads are obtained? Suppose that three heads are obtained, what is the probability that the coin chosen is the biased coin? Solution: F 1 : The chosen coin is biased. F 2 = F c 1 : The chosen coin is fair E : Three heads are obtained By total probability formula: P (Three heads are obtained) = P ( E ) = P ( F 1 ) P ( E | F 1 )+ P ( F 2 ) P ( E | F 2 ) = 1 2 × 0 . 8 3 + 1 2 × ( 1 2 ) 3 =0 . 3185 By Bayes’ Formula P (The chosen coin is biased | Three heads are obtained) = P ( F 1 | E ) = P ( F 1 ) P ( E | F 1 ) P ( F 1 ) P ( E | F 1 P ( F 2 ) P ( E | F 2 ) =( 1 2 × 0 . 8 3 ) / 0 . 3185 . 8038 1

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2. Let ( X, Y ) have joint pdf f X,Y ( x, y )= c , for 0 <x< 1, 0 <y<x 2 . The non-zero region of f X,Y ( x, y ) is also shown by the shaded area in the ﬁgure below. 0
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Practice Midterm11 - Midterm 1 Solution Stat416/Math416...

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