Why are the two random variables independent in the analysis of Randomized Selection algorithm in CLRS?
The two random variables $X_k$ and $T(\max(k-1, n-k))$ are independent because in each recursive call to RANDOMIZED-SELECT() we invoke RANDOMIZED-PARTITION(), a pivot is randomly selected, and the choice of the pivot in one recursive call is independent from the choice of the pivot in another recursive call. Remember that in the analysis we assume that the elements are distinct, so that each element in a particular subarray has the same probability of being selected as the pivot. Moreover, simply stated, the value of $\max(k-1,n-k)$ does not depend on $X_k$: it is the same independently of $X_k$ being equal to zero or one.