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Test_2_Solution - STATISTICS 330 Test 2 Name Last Name...

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1 STATISTICS 330 Test 2 March 12, 2009 Name: _____________________ ____________________ Last Name First Name Student #: _____________ userID: _____________ Instructions: 1. This test paper consists of 6 pages including this title page. 2. A basic scientific calculator is allowed. 5. Duration: 80 min Question Marks Available Marks Obtained Q1 9 Q2 5 Q3 10 Q4 9 Total 33
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2 1. Let X and Y be two random variables with a joint pdf < < < < = otherwise y x y Cx y x f , 0 1 0 , 1 0 , ) , ( 2 a. Show that C = 6. [2 marks] . 6 6 / 1 1 2 1 1 1 1 0 2 1 0 1 0 2 1 0 1 0 2 = = = = = ∫∫ dx x dx ydy x ydxdy x C b. Determine the marginal pdf of Y. [2 marks] . 1 0 , 2 6 6 ) ( 1 0 2 1 0 2 < < = = = y y dx x y dx y x y f Y c. Are X and Y independent random variables? Why? [2 marks] . 1 0 , 3 6 6 ) ( 2 1 0 2 1 0 2 < < = = = x x ydy x dy y x x f X Since ) ( ) ( ) , ( y f x f y x f Y X = , X and Y are independent. d. Find the probability ). ( Y X P > [3 marks] ∫ ∫ = = = = > 1 0 1 0 4 2 2 1 0 0 2 5 3 3 0 2 6 6 ) ( dx x dx x y x dx ydy x Y X P x
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3 2. Let U ~ UNIF(0,1), and suppose that ) , ( ~ | p n BIN p U X = . Find ) ( X E and ) ( X V . [5 marks] 2 2 1 ] [ ] [ ]] | [ [ ] [ n n p nE np E U X E E X E = = = = =
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