Let $\xi_1,\xi_2$ be two real-valued random variables with joint distribution $P$. The latter gives us marginals $p_1,p_2$ conditional distribution $p_{12},p_{21}$ and the copula $C$. The latter concepts are often used to look into conditional structure of random variables, by getting rid of the information contained in marginals. For example, $p_{12}$ being the distribution of $\xi_2$ given $\xi_1$ has no information about $p_1$ (besides which null sets does the latter have, perhaps).

So I hoped that knowing $p_{12}$ without $p_1$ is enough to find $C$. However, that does not seem to be the case. What is the meaning of it? Perhaps I am missing something here - both conditional distributions and copulas should be somehow free from marginals (one of them in case of conditional distribution), but that does not really happen. Is there a notion similar to copula, but which is uniquely determined by conditional distributions?

  • $\begingroup$ Copula operates on distribution functions. But there's a thing called copula density such that $c(F_1(X_1), F_2(X_2)) = \frac{p(X_1, X_2)}{p(X_1) p(X_2)} = \frac{p(X_1 | X_2)}{p(X_1)}$. This formula, indeed, doesn't depend on $p(X_2)$. $\endgroup$ – Artem Sobolev Nov 30 '15 at 15:40
  • $\begingroup$ @Barmaley.exe: I thought of this one, but that assumes existence of the joint density whereas copula is a more general thing. $\endgroup$ – Ilya Dec 1 '15 at 8:28

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