What are some intuitive ways to figure out a joint distribution of 2 RVs if you know their marginal distributions?

This seems to work on a case by case scenario, but I wanted to see if there's some generic way or thought process I can go through to find the joint distribution of 2 dependent random variables if I know their marginal distributions. In some cases, perhaps you can determine this through conditional probabilities, but what if we have no knowledge of conditional probability?

For example, say $$X_1$$ and $$X_2$$ are uniformly distributed on $$[0,1]$$. Say we define $$Y = \min(X_1, X_2)$$ and $$Z = \max(X_1, X_2)$$. To determine $$P(Y \geq y, Z \leq z)$$, you could draw a picture.

• You can get the proper font and spacing for $\min$ and $\max$ using \min and \max. For operators that don't have a command of their own, you can use \operatorname{name}. – joriki May 6 at 5:54

The joint distribution is not determined by the marginal distributions. For instance, if $$X_1$$ and $$X_2$$ are uniformly distributed on $$[0,1]$$, two extremes would be that $$X_1$$ and $$X_2$$ could be independent, or you could have $$X_1=X_2$$. The joint distribution cannot be determined without further information (such as the conditional probabilities).