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# Test _Sol_ - Midterm Test 1(Solution Question 1 The joint...

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Unformatted text preview: Midterm Test 1 (Solution) Question 1 The joint probability distribution function of two continuous random vari- ables X and Y is f XY ( x,y ) = x + xy for 0 < x < 1 and- 1 < y < 1. i. Determine the marginal probability distribution function of X . ii. Determine the marginal probability distribution function of Y . Most got this part correct. f X ( x ) = Z 1- 1 f XY ( x,y ) dy = 2 x f Y ( y ) = Z 1 f XY ( x,y ) dx = y + 1 2 iii. What is the expected value of X + Y ? This is not too difficult either. E ( X + Y ) = E ( X ) + E ( Y ) = Z 1 xf X ( x ) dx + Z 1- 1 yf Y ( y ) dy = 2 3 + 1 3 = 1 iv. Are the two random variables independent? (a) Yes (b) No Definitely (a), since f XY ( x,y ) = f X ( x ) f Y ( y ). v. Which of the following statements are true? I. P ( X = 0) = 0 II. P ( X > 0) = 1 III. P ( X > 1) = 0 (a) I only (b) I and III (c) None of the above (d) All of the above A large number of students put in (a), but the correct answer is (d)....
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Test _Sol_ - Midterm Test 1(Solution Question 1 The joint...

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