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TL;DR

Conditional Expectation as Best Predictor: A canonical quantitative trading interview question at olympiad difficulty. Commonly asked at Two Sigma, DE Shaw, Citadel.

By Valenke Exam Prep Team·Last updated 2026-06-01
olympiadExpected Value & Variance

Conditional Expectation as Best Predictor

Asked at: Two Sigma, DE Shaw, Citadel

Problem
Prove that E[Y∣X]E[Y \mid X] minimizes the mean squared prediction error: for any function g(X)g(X), show E[(Y−E[Y∣X])2]≤E[(Y−g(X))2]E[(Y - E[Y|X])^2] \leq E[(Y - g(X))^2]. Why does this make conditional expectation the "projection" of YY onto functions of XX?
Related concepts
Conditional ExpectationHilbert Spaces

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