Question 4173: Positive-Mass Value Recovered from a Mean

Original question and givens

S={0,…,2}, F=P(S), masses=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, null, -20] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-239/13. Find that slot’s integer value, written as p/1.

• G001 — S={0,…,2}, F=P(S), masses=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, null, -20] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-239/13. 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(23, 2, 1)

total mass numerator = 26

indexed masses=[23, 2, 1];D=26

probabilities=[23/26, 1/13, 1/26]

known numerator0 := mul(23, -22)

known numerator0 = -506

known numerator2 := mul(1, -20)

known numerator2 = -20

known weighted numerator := sum(-506, -20)

known weighted numerator = -526

target expectation numerator := mul(-239/13, 26)

target expectation numerator = -478

unknown weighted contribution := sub(-478, -526)

unknown weighted contribution = 48

recovered unknown value := div(48, 2)

recovered unknown value = 24

recovered weighted numerator := mul(2, 24)

recovered weighted numerator = 48

complete reconstructed numerator := sum(-526, 48)

complete reconstructed numerator = -478

reconstructed expectation := div(-478, 26)

reconstructed expectation = -239/13

k=1;mass_k=2>0;target=-239/13

unique recovered value=24;substitution mean=-239/13

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

24/1

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.