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TL;DR

Uniform Order Statistics and Spacings: A canonical quantitative trading interview question at intermediate difficulty. Commonly asked at Two Sigma, DE Shaw, Citadel.

By Valenke Exam Prep Team·Last updated 2026-06-01
intermediateRandom Variables & Distributions

Uniform Order Statistics and Spacings

Asked at: Two Sigma, DE Shaw, Citadel

Problem
Let U(1)≤U(2)≤⋯≤U(n)U_{(1)} \leq U_{(2)} \leq \cdots \leq U_{(n)} be the order statistics of nn independent  extUniform(0,1)\ ext{Uniform}(0,1) random variables. Find E[U(k)]E[U_{(k)}] and show that the spacings Di=U(i)−U(i−1)D_i = U_{(i)} - U_{(i-1)} (with U(0)=0U_{(0)} = 0) are exchangeable with E[Di]=1/(n+1)E[D_i] = 1/(n+1) for all ii.
Related concepts
Order StatisticsBeta Distribution

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