Same-law second and centered moment cross-checks

Goal and scope

Compute variance at all theta values and verify centered definitions.

Variance measures random spread, not correctness of estimation.

Givens and symbols

• work

• G1 — Theta={1/4,3/4}, Omega={00,01,10,11}; P_(1/4)=(9,3,3,1)/16 and P_(3/4)=(1,3,3,9)/16.
• G2 — U(x)=(x1+x2)/4 has total table(0,1/4,1/4,1/2) in Omega order, with target g(theta)=theta.

• θ — fixed unknown parameter
• U — one total statistic

Rules and sources

• mko_definition · cited — Keep sampling law, positive-mass support and mean at the same theta. Check Var_theta(T)=E_theta[(T−E_theta[T])^2]=E_theta[T^2]−(E_theta[T])^2 exactly. Zero variance means almost-sure constancy under that law, not equality with theta or unbiasedness. · mko-estimator-variance
• work · prerequisite — Finite laws have nonnegative probabilities summing to one; a statistic depends only on data, and expectation sums all weighted outcomes. Var(U)=E[(U-EU)^2]=E[U²]-(EU)².

Main derivation

P_{1/4}^U=(9,6,1)/16,\quad P_{3/4}^U=(1,6,9)/16

E_{1/4}U=1/8,\quad E_{3/4}U=3/8

E_{1/4}U^2=(1/16)(6/16)+(1/4)(1/16)

E_{1/4}U^2=5/128

\operatorname{Var}_{1/4}U=E_{1/4}U^2-(E_{1/4}U)^2

5/128-(1/8)^2=3/128

E_{1/4}(U-1/8)^2

(9/16)(1/8)^2+(6/16)(1/8)^2+(1/16)(3/8)^2=3/128

E_{3/4}U^2=(1/16)(6/16)+(1/4)(9/16)

E_{3/4}U^2=21/128

\operatorname{Var}_{3/4}U=21/128-(3/8)^2

\operatorname{Var}_{3/4}U=3/128

E_{3/4}(U-3/8)^2

(1/16)(3/8)^2+(6/16)(1/8)^2+(9/16)(1/8)^2=3/128

V(x)\equiv0,\quad\theta\in\{1/4,3/4\}

\operatorname{Var}_\theta(V)=0,\quad E_\theta[V]\ne\theta

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, G2, S07, work — Zero variance only gives a.s. constancy under that law.
• S07 · premise_audit · G1, G2, S06, work — Every summand is nonnegative, so a negative variance signals error.
• S06 · premise_audit · G1, G2, S05, work — Equal variance does not imply equal sampling laws.
• S05 · premise_audit · G1, G2, S04, work — Do not import first-law weights.
• S04 · premise_audit · G1, G2, S03, work — The zero support value does not have zero centered deviation.
• S03 · premise_audit · G1, G2, S02, work — Do not subtract(3/8) squared here.
• S02 · premise_audit · G1, G2, S01, work — The squared middle value is 1/16.
• S01 · premise_audit · G1, G2, work — One U may have different centers under different laws.

Error controls and negative controls

Same-law center

Pair theta=1/4 with 1/8 and theta=3/4 with 3/8.

Two formulas

Independently compare centered sums and raw-moment subtraction at 3/128.

• Use the second mean 3/8 for the first variance. → reject — The first law has mean 1/8; centers must match laws.
• Zero variance implies unbiasedness. → reject — V identically0 has zero variance but is biased for theta>0.
• Equal variances imply equal laws. → reject — The U=0 masses differ:9/16 and 1/16.

Result and limitations

Both theta variances are 3/128, verified by centered and raw second moments.

Verification and source binding

3 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 2ad002b889b15f44b5c24941287a4fa911adb4a34398f427dbc154661eb2827b
• original source SHA-256: f2bbc9ab007672ceca721f29145874a8a606c1b10236341756c3630da64cd582
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)