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h4 - Statistics 5101 Fall 2010 Homework Assignment 4 Solve...

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Statistics 5101, Fall 2010: Homework Assignment 4 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. 4-1. If U , V , X , and Y are any random variables, show that cov( U + V, X + Y ) = cov( U, X ) + cov( V, X ) + cov( U, Y ) + cov( V, Y ) 4-2. Suppose X 1 , X 2 , X 3 are IID with mean μ and variance σ 2 . Calculate the mean vector and variance matrix of the random vector Y = Y 1 Y 2 Y 3 = X 1 - X 2 X 2 - X 3 X 3 - X 1 4-3. Suppose X and Y are independent random variables, with means μ X and μ Y , respectively, and variances σ 2 X and σ 2 Y , respectively. Calculate E ( X 2 Y 2 ) in terms of μ X , μ Y , σ 2 X , and σ 2 Y . 4-4. Suppose 6 balls that are indistinguishable except for color are placed in an urn and suppose 3 balls are red and 3 are white. Suppose 2 balls are drawn. What is the probability the one is red and the other white under each of the following conditions?
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