Practice Midterm12

Practice Midterm12 - Practice Midterm 2 Solution J Li Stat...

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Practice Midterm 2 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. The inverse of a 2 × 2 matrix is ± ab cd ² - 1 = ± d ad - bc - b ad - bc - c ad - bc a ad - bc ² 1. The joint density of X and Y is f X,Y ( x, y )= x 2 2 e - xy 0 <x< 2 ,y > 0 . (a) (10) Find the marginal distribution of X , f X ( x ). (b) (10) Find the conditional distribution of Y given X = 1 2 , i.e., f Y | X ( y | 1 2 ). (c) (5) Find E ( Y | X = 1 2 ). (d) (10) Find the probability P ( Y> 1 | X = 1 2 ). Solution: (a) f X ( x Z -∞ f X,Y ( x, y ) dy = Z 0 x 2 2 e - xy dy = x 2 2 · ( - e - xy x ) | 0 = x 2 0 2 (b) f Y | X ( y | 1 2 f X,Y ( 1 2 ) f X ( 1 2 ) = ( 1 2 ) 2 2 e - 1 2 y 1 4 = 1 2 e - 1 2 y y> 0 (c) From part (b), it is shown that Y | X = 1 2 follows the exponential distribution with parameter λ =1 / 2. Hence E ( Y | X = 1 2 )=1 =2 . (d) P ( 1 | X = 1 2 R 1 1 2 e - 1 2 y dy = e - 1 2 . 1
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2. A 4-state Markov chain { X 0 ,X 1 , ..., X n , ... } has a transition probability matrix as follows. The states are 0, 1, 2, and 3.
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Practice Midterm12 - Practice Midterm 2 Solution J Li Stat...

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