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TL;DR

Eve's Law (Total Variance Decomposition): A canonical quantitative trading interview question at intermediate difficulty. Commonly asked at Two Sigma, Citadel, DE Shaw, Jump Trading.

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

Eve's Law (Total Variance Decomposition)

Asked at: Two Sigma, Citadel, DE Shaw, Jump Trading

Problem
Prove the law of total variance:  extVar(Y)=E[ extVar(Y∣X)]+ extVar(E[Y∣X])\ ext{Var}(Y) = E[\ ext{Var}(Y \mid X)] + \ ext{Var}(E[Y \mid X]). Then apply it: if N∼ extPoisson(λ)N \sim \ ext{Poisson}(\lambda) and given N=nN=n, S=X1+⋯+XnS = X_1 + \cdots + X_n where Xi∼ extExponential(1)X_i \sim \ ext{Exponential}(1) i.i.d., find  extVar(S)\ ext{Var}(S).
Related concepts
Law Of Total VarianceCompound Distributions

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