Practice Midterm10

# Practice Midterm10 - Practice Midterm 1 Solution J. Li Stat...

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Practice Midterm 1 Solution J. Li Stat 416, Spring 2005 The number of points assigned to each question is indicated before the question. The full score for the exam is 100. 1. (20) An urn contains 5 black balls and 3 white balls. A ball is randomly drawn from the urn. If it is white, it is put back into the urn. Otherwise, it is not returned to the urn. A second ball is then randomly drawn. What is the probability that the second ball is white? Given that the second ball is white, what is the probability that the ﬁrst ball drawn is also white? Solution: F 1 : The ﬁrst ball is white, F 2 = F c 1 : The ﬁrst ball is black E : The second ball is white By total probability formula: P (The second ball is white) = P ( E ) = P ( F 1 ) P ( E | F 1 )+ P ( F 2 ) P ( E | F 2 ) = 3 8 × 3 8 + 5 8 × 3 7 =0 . 4085 By Bayes’ Formula P (The ﬁrst ball is white | The second ball is white) = 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 ) =( 3 8 × 3 8 ) / 0 . 4085 . 3443 1

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2. Let ( X, Y ) have the joint pdf f X,Y ( x, y )= c 0 <x< 1 , 0 <y< 2 3 + 2 3 x.
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## This note was uploaded on 04/08/2008 for the course STAT 8553 taught by Professor Staff during the Spring '08 term at Columbia.

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Practice Midterm10 - Practice Midterm 1 Solution J. Li Stat...

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