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.


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