Expected number of uniform variates which are strictly increasing

Tags: math

Let (r₁, r₂, …) be an infinite sequence of iid uniform random variates. Let K be the largest number such that r₁ < r₂ < ⋯ < r_K. What is 𝔼[K]?

Note that:

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

We have K ≥ k iff the k elements satisify r₁ < r₂ < ⋯ < r_k, which happens with probability 1/k!. Therefore:

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