Question 4155: Positive-Mass Value Recovered from a Mean

Original question and givens

S={0,…,2}, F=P(S), masses=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, null, -2] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-5/4. Find that slot’s integer value, written as p/1.

• G001 — S={0,…,2}, F=P(S), masses=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, null, -2] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-5/4. Find that slot’s integer value, written as p/1.

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-expectation

Step-by-step derivation

total mass numerator := sum(5, 2, 1)

total mass numerator = 8

indexed masses=[5, 2, 1];D=8

probabilities=[5/8, 1/4, 1/8]

known numerator0 := mul(5, -4)

known numerator0 = -20

known numerator2 := mul(1, -2)

known numerator2 = -2

known weighted numerator := sum(-20, -2)

known weighted numerator = -22

target expectation numerator := mul(-5/4, 8)

target expectation numerator = -10

unknown weighted contribution := sub(-10, -22)

unknown weighted contribution = 12

recovered unknown value := div(12, 2)

recovered unknown value = 6

recovered weighted numerator := mul(2, 6)

recovered weighted numerator = 12

complete reconstructed numerator := sum(-22, 12)

complete reconstructed numerator = -10

reconstructed expectation := div(-10, 8)

reconstructed expectation = -5/4

k=1;mass_k=2>0;target=-5/4

unique recovered value=6;substitution mean=-5/4

Reverse checks

• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute reconstructed expectation with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute complete reconstructed numerator with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute recovered weighted numerator with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute recovered unknown value with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute unknown weighted contribution with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute target expectation numerator with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute known weighted numerator with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute known numerator2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute known numerator0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute total mass numerator with exact rational arithmetic, not rounded decimals.

Result

6/1

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.