ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Proportional stratification reduces variance for a uniform mean

Example

For $P = Unif [0, 1]$ and $f (x) = x$ , split into the two half-intervals and draw one point independently from each. This is proportional allocation with total sample size two and gives variance $1 / 96$ , compared with $1 / 24$ for two ordinary IID draws.

Facts & Assumptions

Given: $A_{1} = [0, 1 / 2)$ , $A_{2} = [1 / 2, 1]$ , $p_{1} = p_{2} = 1 / 2$ , and $n_{1} = n_{2} = 1$ .

[F1]

Stratified Monte Carlo estimator: ${\hat{μ}}_{s t r a t} = p_{1} {\overset{ˉ}{f}}_{1} + p_{2} {\overset{ˉ}{f}}_{2}$ .

[F2]

Stratified estimator unbiasedness, variance, and proportional allocation: The fixed-stratum variance is $\sum_{h} p_{h}^{2} σ_{h}^{2} / n_{h}$ and proportional allocation compares it with the ordinary IID mean of the same total size.

Verification

Given: Independent uniform draws from each half-interval and one ordinary IID comparator sample of size two.

1.1F1F2givenalgebra

Each conditional uniform law has mean $1 / 4$ or $3 / 4$ and variance $(1 / 2)^{2} / 12 = 1 / 48$ . Substitution into the estimator definition [F1] and variance formula [F2] gives $Var ({\hat{μ}}_{s t r a t}) = 2 (1 / 4) (1 / 48) = 1 / 96$ .

2.1givenstep 1.1algebra∎

The ordinary sample mean has variance $Var (U) / 2 = (1 / 12) / 2 = 1 / 24$ . Since $1 / 96 < 1 / 24$ , this specified proportional allocation reduces variance by a factor of four.

Depends on

Used by

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Dependency tree · two levels

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