Question 20761: Finite Mean and Variance Do Not Need a Nonnegative Original Variable

Original question and givens

Use complete normalized finite law masses=[1/16, 2/16, 13/16] and aligned values X=[0, 11, 14]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).

• G001 — Use complete normalized finite law masses=[1/16, 2/16, 13/16] and aligned values X=[0, 11, 14]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-chebyshev-inequality

Step-by-step derivation

source operation=cheb-qualification;original snapshot and mathematics retained

explicit ordered source output fields=[masses, values, mean, variance, finite_moment_certificate, certificate]

Chebyshev complete probability law mass := sum(1/16, 1/8, 13/16)

Chebyshev complete probability law mass = 1

Chebyshev original atom0 weighted SIGNED X := mul(1/16, 0)

Chebyshev original atom0 weighted SIGNED X = 0

Chebyshev original atom1 weighted SIGNED X := mul(1/8, 11)

Chebyshev original atom1 weighted SIGNED X = 11/8

Chebyshev original atom2 weighted SIGNED X := mul(13/16, 14)

Chebyshev original atom2 weighted SIGNED X = 91/8

Chebyshev complete SAME-law mean := sum(0, 11/8, 91/8)

Chebyshev complete SAME-law mean = 51/4

Chebyshev original atom0 X minus SAME mean := sub(0, 51/4)

Chebyshev original atom0 X minus SAME mean = -51/4

Chebyshev original atom0 centered SQUARE := mul(-51/4, -51/4)

Chebyshev original atom0 centered SQUARE = 2601/16

Chebyshev original atom0 weighted centered square := mul(1/16, 2601/16)

Chebyshev original atom0 weighted centered square = 2601/256

Chebyshev original atom1 X minus SAME mean := sub(11, 51/4)

Chebyshev original atom1 X minus SAME mean = -7/4

Chebyshev original atom1 centered SQUARE := mul(-7/4, -7/4)

Chebyshev original atom1 centered SQUARE = 49/16

Chebyshev original atom1 weighted centered square := mul(1/8, 49/16)

Chebyshev original atom1 weighted centered square = 49/128

Chebyshev original atom2 X minus SAME mean := sub(14, 51/4)

Chebyshev original atom2 X minus SAME mean = 5/4

Chebyshev original atom2 centered SQUARE := mul(5/4, 5/4)

Chebyshev original atom2 centered SQUARE = 25/16

Chebyshev original atom2 weighted centered square := mul(13/16, 25/16)

Chebyshev original atom2 weighted centered square = 325/256

Chebyshev full variance E centered square := sum(2601/256, 49/128, 325/256)

Chebyshev full variance E centered square = 189/16

ORIGINAL complete normalized masses=[1/16, 1/8, 13/16];ALL aligned SIGNED values X=[0, 11, 14];SAME-law mean=51/4;all deviations=[-51/4, -7/4, 5/4]

ALL nonnegative squared deviations Y=[2601/16, 49/16, 25/16];variance=E[Y]=189/16;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms

Reverse checks

• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Recompute Chebyshev full variance E centered square with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Chebyshev original atom2 weighted centered square with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Chebyshev original atom2 centered SQUARE with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Chebyshev original atom2 X minus SAME mean with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Chebyshev original atom1 weighted centered square with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Chebyshev original atom1 centered SQUARE with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Chebyshev original atom1 X minus SAME mean with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Chebyshev original atom0 weighted centered square with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Chebyshev original atom0 centered SQUARE with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Chebyshev original atom0 X minus SAME mean with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Chebyshev complete SAME-law mean with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Chebyshev original atom2 weighted SIGNED X with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Chebyshev original atom1 weighted SIGNED X with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Chebyshev original atom0 weighted SIGNED X with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Chebyshev complete probability law mass with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([1/16, 1/8, 13/16], [0, 11, 14], 51/4, 189/16, Yes, finite_mean_and_variance)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.