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Unformatted text preview: iμ 1 , 1 ) 2 ( 1 n P n j =1 X 2 j Y 2 j )( 1 n P n k =1 X k Y k ) 2 . (Hint: Use Slutsky’s theorem.) (c) Find the approximate distribution of ( ¯ X n )( ¯ Y n ). (Hint: Use the multivariate delta method.) 3. (a) Let X and Y be independent continuous random variables each of them having a density belonging to an exponential family. Prove or disprove: The random vector ( X, Y ) has a joint density belonging to an exponential family. (b) Let ( X, Y ) be jointly continuous random variables having a joint density belonging to an exponential family. Prove or disprove: The marginal distribution of X has a density belonging to an exponential family. (Hint: Consider a joint density in the form g ( θ ) e θxy I (0 , 1) ( x ) I (0 , 1) ( y ). )...
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 Spring '08
 Hannig,J
 Statistics, Variance, Probability distribution, Probability theory, probability density function, exponential family

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