How would you evaluate this if X1, X2 are independent exponentials?
I know what $min(X1,X2)$ is and what $max(X1,X2)$ is. However, while $min(X1,X2)$ is exponential with parameter (lambda1 + lambda2) -- the max is not.
Is the only way to proceed is to say:
1) Suppose $X_1 > X_2$. Then I find $Pr(R <= r) = Pr(X_1 - X_2 <= r)$ by doing a double integral. 2) Suppose $X_2 <X_1$. Then I find $Pr(R <= r) = Pr(X_2 - X_1 <= r)$ by doing a double integral. 3) Calculate the CDF as:
$Pr(X_1 - X_2 <= r) Pr(X_2 <= X_1) + Pr(X_2 - X_1 <= r)*Pr(X_1 <= X_2)$.
That should give me $Pr(R \leq r)$. I then would then differentiate with respect to r to get $F_R(r)$.
Is there perhaps a more clever way to do it?