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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.

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    $\begingroup$ 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}. $\endgroup$ – joriki May 6 at 5:54
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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).

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