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Suppose I have two independent random varaibles X and Y if I add random variable Z which is correlated to the both of X and Y, Will they be independent anymore?

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Probably not, but this is not a suitable question for MO (textbook question, see the faq). – BenoĆ®t Kloeckner Jul 21 2010 at 9:35

closed as too localized by Gjergji Zaimi, S. Carnahan Jul 21 2010 at 20:10

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