Variance in finite probability models

Mathematical statement

Put μ=E[X]. In a valid finite probability model, define V(X)=Var(X)=Σ_s w_s(X(s)−μ)²≥0. Translation leaves variance unchanged and Var(kX)=k²Var(X). Variance is zero exactly when X has one common value at every positive-weight outcome.

Representative formula (exact TeX)

V(X)

Assumptions

• The finite nonempty sample space, normalized nonnegative weights, and total real-valued function are explicit.
• Use probability-distribution variance rather than a sample estimator, distinguishing positive-weight support from the whole sample space.

Symbol scope

• V — variance of a finite probability distribution
• X — real-valued random variable

Intuition, definition, and misconceptions

Every weighted squared term is nonnegative. Zero-weight outcomes may have different values without affecting the moments. Zero variance means constant almost everywhere; in this finite model, that is constant on the positive-weight support.

For X+c, the mean also increases by c, leaving deviations unchanged. For kX, deviations are multiplied by k and squaring gives k². Keep the cases k=0, negative k, and one-point support explicit.

The exercises compute exact rational moments without floating approximations of E[X²]−E[X]². They do not use an n−1 sample denominator or unconditionally split Var(X+Y) into two variances. Standard deviation and general covariance have separate obligations.

Common errors include requiring equal values even at zero-probability outcomes for zero variance, or confusing sample variance with distribution variance or standard deviation.

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

• Expectation in finite models — Variance first requires the expectation.
• Natural-number powers and principal square roots — Squared deviations and k² use power semantics.

Verification boundary

Finite central moments can be recomputed exactly; general variance, almost-everywhere equivalence, and integrability have no attached formal Evidence.

not_formalized

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

Next obligation

Bind finite centered moments, constancy on positive support, and the bridge to ProbabilityTheory.variance.

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.