Midterm1-2010

# A what is p x 0 1 point b what is p 0 x 1 2 points c

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0 is an unspecified parameter, and e is the usual mathematical constant. (a) What is P ( X 0)? ( 1 point. ) (b) What is P (0 X 1)? ( 2 points. ) (c) What is E ( X )? ( 2 points. ) Write your answer to Question 1 in the space below. 2

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Question 2 Suppose X and Y are continuous random variables with joint pdf f XY given by f XY ( x, y ) = 3 2 ( x 2 + y 2 ) for 0 x 1 and 0 y 1 0 otherwise. (a) What are the marginal pdfs of X and Y ? ( 1 point. ) (b) What is Cov( X, Y )? ( 2 points. ) (c) What is E ( Y X = 1 2 ) ? ( 2 points. ) Write your answer to Question 2 in the space below. 5

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Question 3 Suppose X is a continuous random variable with the standard normal distribution: X N (0 , 1). We know from statistical tables that P ( | X | ≤ 1 . 96) = 0 . 95, to two decimal places. But suppose we did not have any statistical tables to consult, and we could not remember this fact. Use Chebyshev’s inequality to prove that P ( | X | ≤ 1 . 96) 0 . 73. Hint: 1 - (1 . 96) - 2 = 0 . 74, to two decimal places. ( 5 points. ) Write your answer to Question 3 in the space below. 8

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Question 4 Suppose that X 1 , X 2 , . . . are independent random variables, each of which is equal to - 1 with probability 1 2 , and equal to 1 with probability 1 2 . Let Z n denote the quantity Z n = 1 n n X i =1 X i . (a) What are E ( X i ) and Var( X i )? ( 1 point. ) (b) What is P ( | Z 4 | > 1 . 96)? ( 2 points. ) (c) What is lim n →∞ P ( | Z n | > 1 . 96)? ( 2 points. ) Write your answer to Question 4 in the space below. 10

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