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h6 - Stat 5101(Geyer Fall 2011 Homework Assignment 6 Due...

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Stat 5101 (Geyer) Fall 2011 Homework Assignment 6 Due Wednesday, October 26, 2011 Solve each problem. Explain your reasoning. No credit for answers with no explanation. If the problem is a proof, then you need words as well as formulas. Explain why your formulas follow one from another. 6-1. The function f ( x ) = 2 x, 0 < x < 1 is a PDF. Suppose X is a random variable having this PDF. (a) Calculate E ( X ). (b) Calculate E ( X 2 ). (c) Calculate var( X ). 6-2. The function f ( x, y ) = x + y, 0 < x < 1 , 0 < y < 1 is a PDF. Suppose ( X, Y ) is a random vector having this PDF. (a) Calculate E ( X ). (b) Calculate var( X ). (c) Calculate cov( X, Y ). (By symmetry, E ( Y ) = E ( X ) and var( Y ) = var( X ) so we do not need to calculate them.) 6-3. Suppose ( X, Y ) is a continuous random vector having PDF f. Say for each of the following definitions of f whether X and Y are independent or not. (a) f ( x, y ) = 4 xy , 0 < x < 1, 0 < y < 1. (b) f ( x, y ) = 8 xy , 0 < x < y < 1. (c) f ( x, y ) = 144( x - 1 / 2) 2 ( y - 1 / 2) 2 , 0 < x < 1, 0 < y < 1. (d) f ( x, y ) = 288( x - 1 / 2) 2 ( y - 1 / 2) 2 , 0 < x < y < 1. 1
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6-4. Suppose X is a continuous random variable having PDF f ( x ) = 1 + x, - 1 x < 0 1 - x, 0 x 1 0 ,
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