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My question is probably very basic. Let's consider random variable from this answer. It looks like probability mass function for each $\xi_i$ is 1/3, 1/3, 1/3 at points 1,2,3. So given pdf for these random variables we cannot find for example $P(\xi_1 < \xi_2)$. So pdf does not contain all info about random variable ? Is there an error in my reasoning?

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up vote 2 down vote accepted

The individual (marginal) pdfs for each random variable, as you have noted, do not contain the information needed to calculate $\mathbb P(\xi_1 < \xi_2)$. But the joint pdf, which you can figure out from the definitions of those RVs as given in that answer, does contain all the information about their distributions.

Marginals alone aren't enough, but the joint pdf is.

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