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If two random variables $X$ and $Y$ are independent, are $X$ and $Y$ conditionally independent given any choice of a random variable $Z$?

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Remember that independence between two random variables means that the occurrence of one variable does not affect the probability of occurrence of the other. Conditional independence, however, relates to the independence of two variables when a third variable is given. Think about whether the introduction of random variable Z could introduce a dependency between X and Y. Don't assume conditional independence just because X and Y are independently distributed; consider whether knowing Z gives any additional information about the relationship between X and Y.

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