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Suppose that a distribution F is given. Does there always exist a sequence of independent random variables following F? Moreover, given two distributions F and G and a random variable X following F, does there exist a random variable Y following G, such that X and Y are independent?

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    @Ross Millikan I am sure the OP means "independent random variables having same CDF $F$"2012-10-18

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The answer to the first question is YES by a consequence of the famous Kolmogorov extension theorem. Answer to the second is NO. See the comments below.

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    I've edited the answer accordin$g$ly. Thank you.2012-10-18