Uniform Draws and Sums

What this uniform-sum stopping-time question tests

This is a medium-difficulty probability question that combines uniform random variables, partial sums, and stopping times—a classic setup in probability interviews at quantitative trading and finance firms. It rewards clean reasoning about expectations under a stopping rule, not computational brute force.

To solve it, candidates must identify the stopping rule (the first time a sum exceeds a threshold), set up the problem correctly in terms of a random stopping time, and then apply a key insight or technique to compute the expectation. The approach typically involves either a clever symmetry argument, a recursive or martingale-based perspective, or a direct integral calculation. Interviewers value a structured setup and clear justification of your method.

• Properties of uniform random variables and their sums
• Stopping times and optional stopping
• Linearity of expectation under stopping rules
• Connection to Poisson processes and order statistics