Law of total variance

Mathematical statement

Var(X)=E[Var(X|G)]+Var(E[X|G]). Partition cells are pairwise disjoint, nonempty and cover every outcome; a zero-mass cell is still a nonempty event consisting of zero-mass outcomes. Cell means and variances use the same law, with division only on positive-mass cells. Null-cell conditional versions are determined only almost surely and their outer weighted contribution is0. Neither term may be omitted or have its meaning exchanged. No independence of partition events is required.

Representative formula (exact TeX)

V(X)=E(V(X\mid G))+V(E(X\mid G))

Assumptions

• The complete normalized finite law, exact values and complete disjoint partition of nonempty cells are explicit.
• G is generated by the partition; do not perform conditional division on zero-mass cells.

Symbol scope

• V — variance operator
• X — finite random variable
• E — expectation operator
• G — partition-generated sigma-algebra

Intuition, definition, and misconceptions

Cells are nonempty; zero mass means the total probability of their outcomes is0.

The within term averages cell variances.

The between term is the variance of cell means; null cells affect neither outer weighted term.

Do not use an empty cell as a zero-mass event, omit the between term, swap term meanings or divide by0.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Law of total expectation — Use the tower identity.
• Variance in finite probability models — Global and cell variances are required.
• Finite conditional expectation — Use the cell means.

Verification boundary

This slice uses complete finite laws; general conditional variance needs a measure-theoretic proof.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add bias-variance decomposition.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.