ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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.

An all-zero Monte Carlo sample can report zero standard error

Example

Let $n \geq 2$ and $p \in (0, 1)$ . On the product probability space ${0, 1}^{n}$ with independent Bernoulli $(p)$ coordinates $X_{1}, \dots, X_{n}$ , define ${\overset{ˉ}{X}}_{n} : = n^{- 1} \sum_{j = 1}^{n} X_{j}$ , $s_{n}^{2} : = (n - 1)^{- 1} \sum_{j = 1}^{n} (X_{j} - {\overset{ˉ}{X}}_{n})^{2}$ , and ${\hat{s e}}_{n} : = s_{n} / \sqrt{n}$ . The event that all coordinates are zero has probability $(1 - p)^{n} > 0$ . On that event ${\overset{ˉ}{X}}_{n} = s_{n}^{2} = {\hat{s e}}_{n} = 0$ , although the target mean is $p > 0$ and the population variance is $p (1 - p) > 0$ , so the absolute error is $p$ . This is an exact finite-sample probability-model example, not an experimental observation; it shows that a zero plug-in standard error in one realization does not establish zero population variance or zero estimation error.

Example references

IID sample mean, variance, and plug-in standard error definitions

Population variance definition

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 2, §2.2

Facts & Assumptions

Given: $n \geq 2$ , $p \in (0, 1)$ , and $X_{1}, \dots, X_{n}$ the coordinate random variables on the stated Bernoulli product probability space.

[F1]

Identical distribution and IID families: IID means common law and mutual independence.

[F2]

IID Monte Carlo sample mean, variance, and standard error: ${\overset{ˉ}{X}}_{n} = n^{- 1} \sum_{j = 1}^{n} X_{j}$ , $s_{n}^{2} = (n - 1)^{- 1} \sum_{j = 1}^{n} (X_{j} - {\overset{ˉ}{X}}_{n})^{2}$ , and ${\hat{s e}}_{n} = s_{n} / \sqrt{n}$ for $n \geq 2$ .

[F3]

Moments, variance, and covariance on a probability space: for square-integrable $X$ , $Var (X) = E [(X - E [X])^{2}]$ .

Verification

1.1F1givenalgebra

Mutual independence gives $P (X_{1} = \dots = X_{n} = 0) = \prod_{j = 1}^{n} P (X_{j} = 0) = (1 - p)^{n} > 0$ .

2.1F2givenstep 1.1

On this event all $X_{j}$ and ${\overset{ˉ}{X}}_{n}$ are zero, so [F2] gives $s_{n}^{2} = 0$ and ${\hat{s e}}_{n} = 0$ .

3.1F3givenstep 1.1algebra∎

For a Bernoulli $(p)$ variable $X_{j}$ , $X_{j}^{2} = X_{j}$ pointwise, hence $E [X_{j}] = p$ and [F3] gives $Var (X_{j}) = E [X_{j}^{2}] - p^{2} = p (1 - p) > 0$ . On the event in step 1.1 the sample mean differs from this target by $p$ .

Depends on

Dependency tree · 2 levels

Sources

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 2, §2.2