Let $X_1, X_2, \dots, X_n \sim U(0, 1)$ be $n$ random variables distributed uniformally in $(0, 1)$. Let $X_{(k)}$ be the $k$-th ordered statistics, i.e. $$X_{(1)} < X_{(2)} < \dots < X_{(n - 1)} < X_{(n)}$$
How one can find the distribution function for $X_{(i)} - X_{(j)}$ (assume that $i > j$), i.e. the probability $\Bbb{P}(X_{(i)} - X_{(j)} < t)$, where $t \in (0, 1)$?
Perhaps, the straightforward approach would be to compute the joint distribution of $X_{(i)}$ and $X_{(j)}$ first. However, it also seems to be not so trivial. I also hope that there may be some trick I'm not aware of to solve the initial problem without computing the joint distribution. It would be absorbing to me to see different approaches to this problem.
I'm also interested in particular case when $j = i + 1$.