Expected number of uniform variates which are strictly increasing

Tags: math

Let (r_1, r_2, \ldots) be an infinite sequence of iid uniform random variates. Let K be the largest number such that r_1 < r_2 < \cdots < r_K. What is \mathbb{E}[K]?

Note that:

\mathbb{E}[K] = \sum_{k=1}^{\infty} \text{Pr}[K \ge k].

We have K \ge k iff the k elements satisify r_1 < r_2 < \cdots < r_k, which happens with probability 1/k!. Therefore:

\mathbb{E}[K] = \sum_{k=1}^{\infty} \frac1{k!} = e - 1.