Hard questions require advanced understanding and critical thinking. Here, your problem-solving skills are key, as these questions often involve complex scenarios needing in-depth analysis and well-structured responses.

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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