Practice Midterm13

# Practice Midterm13 - Midterm 2 Solution J. Li Stat 416,...

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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 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, i.e., f Y | X ( y | 1). (c) (10) Find the probability P ( Y< 2 | X =1). (d) (10) Find E ( Y 2 +3 Y | X =1) . Solution: (a) f X ( x Z -∞ f X,Y ( x, y ) dy = Z 0 x 2 e - xy dy = x 2 · ( - e - xy x ) | 0 = 1 2 0 2 (b) f Y | X ( y | 1) = f X,Y (1 ) f X (1) = 1 2 e - y 1 2 = e - y y> 0 (c) P ( 2 | X =1)= R 2 0 e - y dy =1 - e - 2 . (d) From part (b), it is shown that Y | X = 1 follows the exponential distribution with parameter λ = 1. Hence E ( Y 2 Y | X E ( Y 2 | X =1)+3 E ( Y | X 2 2 =5. 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. 3 4 1 4 00 1 4 3 4 1 2 0 1 4 1 4 0 1 2 1 4 1 4 (a) (10) Specify the classes of the Markov chain.
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## This note was uploaded on 04/08/2008 for the course STAT/MATH 416 taught by Professor Li during the Spring '08 term at Columbia.

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Practice Midterm13 - Midterm 2 Solution J. Li Stat 416,...

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