[math-fun] Permutation question

Let P be a permutation of {1, 2, ..., n}. Let the longest initial decreasing prefix of P have k elements. For example, let n = 5 and P = (5, 3, 2, 4, 1) Then the longest decreasing prefix of P is (5, 3, 2), so k = 3 and P_k = 2. In terms of n, what is the expected value of k? of P_k?