test2 - Problem 2. (4 points) Let ( X, Y ) be the...

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MAT 521 Test 2 June 15, 2004 Instructions: Write the answers and show all your work in the blue books. There are 4 problems. Make sure you do all 4. No books, notes, or collaboration with others. Problem 1. (10 points) An urn contains 20 balls: 5 white, 5 black, and the rest various other colors. Let X and Y be the numbers of white and black balls, respectively, in a sample of size n = 4 . Let f ( x, y ) be the joint probability function of X and Y . a. Find f (1 , 2) if the sampling is without replacement. b. If sampling is with replacement then the marginal distributions of X and Y are binomial. What are the appropriate values of the binomial parameters n and p ? c. Find P ( Y = 2) if the sampling is with replacement.
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Unformatted text preview: Problem 2. (4 points) Let ( X, Y ) be the coordinates of a point picked at random from a triangular region. The triangle is the one with vertices at (0,0), (0,2), and (3,2). Find P ( Y > X ) . Problem 3. (6 points) Let X and Y be independent and indentically dis-tributed with common density f ( x ) = 2 x, < x < 1 . a. What is the joint density function of the pair ( X, Y )? b. Find P ( X + Y > 1) . Problem 4. (10 points) Let X and Y have joint density f ( x, y ) = c xy, ≤ x ≤ y ≤ 1 . a. Find the value of the constant c . b. Find the marginal density of X . c. Are X and Y independent? Briefly justify. 1...
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This note was uploaded on 02/17/2011 for the course MAT 521 taught by Professor Staff during the Spring '08 term at Syracuse.

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