Question: Given a triple \(\left(X_{1}, X_{2}, X_{3}\right)\), we can define 3 p.m.'s \(\mu_{12}, \mu_{13}, \mu_{23}\) on \(\mathbb{R}^{2}\) by \(\mu_{i j}\) is the distribution of \(\left(X_{i}, X_{j}\right)\)

These p.m.'s satisfy a consistency condition:

the marginal distribution \(\mu_{1}\) obtained from \(\mu_{12}\) must coincide with the marginal obtained from \(\mu_{13}\), and similarly for \(\mu_{2}\) and \(\mu_{3}\).

Give an example to show that the converse is false. That is, give an example of \(\mu_{12}, \mu_{13}, \mu_{23}\) satisfying (2) but for which there does not exist a triple \(\left(X_{1}, X_{2}, X_{3}\right)\) satisfying (1)

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