Exercise 7.4.4

Show that RANDOMIZED-QUICKSORT's expected running time is $\Omega(n\lg{n})$.

We use the same reasoning for the expected number of comparisons, we just take in in a different direction.

$$ \begin{align} \E[X] &= \sum_{i=1}^{n-1} \sum_{j=i+1}^n \frac{2}{j-i+1} \\ &= \sum_{i=1}^{n-1} \sum_{k=1}^{n-i} \frac{2}{k + 1} & (k \ge 1) \\ &\ge \sum_{i=1}^{n-1} \sum_{k=1}^{n-i} \frac{2}{2k} \\ &\ge \sum_{i=1}^{n-1} \Omega(\lg{n}) \\ &= \Omega(n\lg{n}) \end{align} $$