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Unformatted text preview: (b) Are X and Y statistically independent? Justify your answer. 1 Problem 3 At a party n people put their hats in the center of a room, where the hats are mixed together. Each person then randomly selects one with equal probability. Let us deﬁne ~ X = [ X 1 X 2 ··· X n ], where X i = ( 1 , if the i th person selects his or her own hat , otherwise. (a) Find E [ X i ] and var( X i ). (b) Find the covariance matrix, C ~ X . For n = 3, ﬁnd its eigenvalues and eigenvectors. Write it in the form C ~ X = V Λ V T , where Λ is a diagonal matrix. (c) Let S = X 1 + X 2 + ··· + X n . Find E [ S ] and σ 2 S . 2...
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 Spring '08
 Staff
 Variance, Probability theory, Random Signal Analysis, marginal densities fX

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