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.