Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10
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Conditional variance decomposition

Statement

Assume AC. For real X∈L2(P), Var⁡(X)=E[Var⁡(X∣G)]+Var⁡(E[X∣G]).

Facts & Assumptions

Given: AC, real X∈L2(P) and a sub-sigma-algebra G.

[F1]

The integrable conditional variance equals the conditional second moment minus the squared conditional mean. (Conditional variance is well-defined and has the second-moment formula)

[F2]

Taking ordinary expectation of a conditional expectation preserves its value. (Basic algebra and order properties of conditional expectation)

[F3]

Variance is the expectation of the centered square. (Moments, variance, and covariance on a probability space)

Proof

technique · direct
1.1

Put U=E[X∣G]. By [F1], U2=E[X2∣G]−Var⁡(X∣G) is a difference of integrable functions, so U is square integrable. Taking expectations in that formula gives E[Var⁡(X∣G)]=EX2−EU2. Moreover EU=EX by [F2].

F1F2
2.1

For any square integrable real V, expansion of the centered square in [F3] gives Var⁡(V)=EV2−(EV)2. Therefore E[Var⁡(X∣G)]+Var⁡(U)=EX2−EU2+EU2−(EX)2=Var⁡(X), proving the formula.

step 1.1F3∎

Source notes

Durrett §4.1.2, printed pp.210–213, supplies expectation preservation and the conditional L2 identities. The decomposition is the displayed local algebraic consequence of the proved second-moment formula.

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