Probability that two elements are compared in randomized quicksort
The idea of the proof is to compute, for any two elements $x,y$ in the array, the probability that they are compared in the algorithm. This probability could potentially depend on the entire array. However, it turns out that you can compute it only given the order statistics of $x,y$, that is, their relative order in the sorted array. If you know that $x$ is the $i$th smallest element in the array and that $y$ is the $j$th smallest element in the array, then the probability that $x,y$ are compared is $\frac{2}{|j-i|+1}$.

This is not a special case – every element $x$ in the array is the $i$th smallest element, for some value of $i$. This is just the pertinent information that allows us to calculate the probability that $x$ and $y$ are compared.