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# Range of Order Statistic

with a exp(1) distribution find pdf of the range

$$X_{(i)} = S$$
$$X_{(n)} = R+S$$

Using joint order statistic

$$f_{ X_{(i)}, X_{(j)} }(u, v) =$$
$$\frac{ n! }{ (i-1)! (j-1-i)! (n-j)! } [ F_X(u) ]^{i-1} [F_X(v) – F_X(u)]^{j-1-i} [1 – F_X(v)]^{n-j} f_X(u) f_X(v)$$

i get

$$f_{R,S}(r,s) =$$
$$\frac{ n! }{(n-2)! } [F_X(s) – F_X()]^{n-2} f_X(u) f_X(v)$$
$${n(n-1)} [1-e^{-r-s} – (1-e^{-s})]^{n-2} e^{-s}e^{r+s}$$

which i get $$n(n-1)e^{-r}e^{-2s}[e^{-r}]$$

but i feel this is the wrong answer to the joint distribution. Am i doing something worng? I know i need to integrate afterwards with respect to s but im not sure if im correct