# Joint distribution of min and absolute difference

Let $$X_1, X_2 \sim Exp(1)$$ be independent random variables. Let $$Y = \min\{X_1,X_2\}, Z = |X_1-X_2|$$. What is joint distribution of $$Y,Z$$?

I think I can figure out the individual distributions of $$Y$$ and of $$Z$$. e.g. for $$Y$$, $$P(Y \leq x) = P(X_1 \leq x \ \lor \ X_2 \leq x) = 1 - P(X_1 > x \ \land \ X_2 > x) = 1 - P(X_1 > x)P(X_2 > x)$$

and for $$Z$$ I can break down $$|X_1 - X_2|$$ into the two cases $$X_1 > X_2$$, $$X_2 > X_1$$. But when it comes to the joint distribution I don't know where to start, because there seems to be dependence between $$Y,Z$$. E.g. if $$Y$$ is large, then both $$X_1, X_2$$ are large, which in my mind would make the probability of a large $$Z$$ value small.

I suspect I may have to use the memorylessness property of the exponential distribution but I don't know how.