# Constructing a joint distribution from pairwise bivariate marginal distributions?

It's fairly well-known that given univariate distribution functions $F_X, F_Y, F_Z$, one can construct the joint distribution $F_{(X, Y, Z)}(x, y, z) = C(F_{X}(x), F_{Y}(y), F_{Z}(z))$, where $C$ is some choice of copula function, which is basically any multivariate distribution with uniform marginals. Different copulas give different joint distributions.

Given bivariate marginals $F_{(X, Y)}, F_{(X, Z)}, F_{(Y,Z)}$ can I do something similar -- by specifying some copula-esque function can I specify one of the many possible joint distributions with those marginals? This question has an answer giving two possible ideas, but I'm wondering if there's a more general way of thinking about it.

The second part of the question here is interesting and gets at the same point as my question. The best answer shows that seemingly reasonable combinations of marginal distributions might be impossible to generate from any joint distribution. But let's assume that's not a problem here, and that the marginals were definitely derived from a real joint distribution, just one that I don't know.

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• I do not really get up your idea. However, are you looking for how many possible joint distribution function that can be constructed from those margins? if yes, did you try pair-copula model instead. – user151245 Jan 24 '18 at 11:55
• Thank you! This is a good approach. While I don't actually need to express the bivariate margins as copulas themselves, it looks like the general idea of combining them using a conditional copula will be useful. – Michael Curry Jan 24 '18 at 16:47