The Secretary Problem

You observe $n$ candidates (i.i.d. continuous draws) one at a time in random order. After seeing each candidate, you only know their relative rank among those seen so far. You must irrevocably accept or reject each candidate on the spot, and you win only if you select the overall best candidate. Consider the threshold strategy: skip the first $r-1$ candidates (the "exploration phase"), then accept the next candidate who is the best seen so far (a "record"). 1. Prove that this class of threshold strategies contains the optimal strategy -- i.e., the strategy maximizing the probability of selecting the global best is of this form. 2. Write the exact success probability $P(r, n)$ as a function of $r$ and $n$. Find the asymptotically optimal $r^{*}$ and the limiting success probability as $n \to \infty$.