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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.

A three-unit SRSWOR variance calculation

Example

Let $U = {1, 2, 3}$ with fixed values $y = (1, 2, 4)$ , and select an SRSWOR sample of size $n = 2$ . The three samples each have probability $1 / 3$ . Their Horvitz–Thompson totals are $9 / 2$ , $15 / 2$ , and $9$ , respectively, so the true design variance of the total is $7 / 2$ . The pair-inclusion and Sen–Yates–Grundy sample estimates are $3 / 4$ , $27 / 4$ , and $3$ ; their mean is also $7 / 2$ . The sample means are $3 / 2$ , $5 / 2$ , and $3$ , with variance $7 / 18$ . The finite support of three values is not an exact normal law.

Sample	$p_{d} (s)$	${\hat{T}}_{H T}$	${\hat{V}}_{H T}$	${\hat{V}}_{S Y G}$	${\overset{ˉ}{y}}_{s}$
${1, 2}$	$1 / 3$	$9 / 2$	$3 / 4$	$3 / 4$	$3 / 2$
${1, 3}$	$1 / 3$	$15 / 2$	$27 / 4$	$27 / 4$	$5 / 2$
${2, 3}$	$1 / 3$	$9$	$3$	$3$	$3$

Facts & Assumptions

Given: $U = {1, 2, 3}$ , $y_{1} = 1, y_{2} = 2, y_{3} = 4$ , and SRSWOR of size two.

[F1]

Simple random sampling without replacement: The support consists of all two-element subsets, each with probability $1 / (\binom{3}{2}) = 1 / 3$ .

[F2]

Horvitz–Thompson variance and pair-inclusion estimate: If $π_{i} > 0$ , the total variance is $\sum_{i, j} Δ_{i j} y_{i} y_{j} / (π_{i} π_{j})$ ; if all $π_{i j} > 0$ , the pair-inclusion estimate has expectation equal to that variance.

[F3]

Sen–Yates–Grundy variance identity: For a fixed-size design, ${Var}_{d} ({\hat{T}}_{H T}) = - \frac{1}{2} \sum_{i, j} Δ_{i j} (y_{i} / π_{i} - y_{j} / π_{j})^{2}$ , and the sample pair-inclusion version is unbiased when every $π_{i j} > 0$ .

[F4]

SRSWOR inclusion covariance and sample-mean variance: For $N > 1$ and $1 \leq n \leq N$ , ${Var}_{d} ({\overset{ˉ}{y}}_{S}) = (1 - n / N) S_{U}^{2} / n$ .

Verification

1.1givenF1choosealgebra

By [F1], the support is ${{1, 2}, {1, 3}, {2, 3}}$ with equal probabilities. Counting occurrences gives $π_{i} = 2 / 3$ , $π_{i j} = 1 / 3$ for $i \neq j$ , $Δ_{i i} = 2 / 9$ , and $Δ_{i j} = - 1 / 9$ for $i \neq j$ ; the population mean is $7 / 3$ and $S_{U}^{2} = ((1 - 7 / 3)^{2} + (2 - 7 / 3)^{2} + (4 - 7 / 3)^{2}) / 2 = 7 / 3$ .

2.1givenF1F2step 1.1algebra

Each observed total is $(3 / 2)$ times the sum of the two observed values, so the HT totals are $9 / 2, 15 / 2, 9$ and have mean $7$ ; their design variance is $\frac{1}{3} ((9 / 2 - 7)^{2} + (15 / 2 - 7)^{2} + (9 - 7)^{2}) = 7 / 2$ .

2.2F1F2F3step 1.1algebra

Put $x_{i} = y_{i} / π_{i}$ , so $(x_{1}, x_{2}, x_{3}) = (3 / 2, 3, 6)$ . For any sampled pair ${i, j}$ , the HT pair estimate and the SYG estimate both equal $\frac{1}{3} (x_{i} - x_{j})^{2}$ , since $Δ_{i i} / π_{i} = 1 / 3$ and $Δ_{i j} / π_{i j} = - 1 / 3$ ; thus they are $3 / 4, 27 / 4, 3$ and their expectation is $\frac{1}{3} (3 / 4 + 27 / 4 + 3) = 7 / 2$ by [F2] and [F3].

3.1F1F4step 1.1algebra∎

The sample means are $3 / 2, 5 / 2, 3$ , with average $7 / 3$ and design variance $\frac{1}{3} ((3 / 2 - 7 / 3)^{2} + (5 / 2 - 7 / 3)^{2} + (3 - 7 / 3)^{2}) = 7 / 18$ ; [F4] gives $(1 - 2 / 3) (7 / 3) / 2 = 7 / 18$ , and the three supported means form a finite set rather than an exact normal calibration.

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