Question 31720: Expectation in a Nonuniform Finite Model

Original question and givens

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -25), (1, -25), (2, -4), (3, 1018)}. In the displayed order of S, the nonnegative probability weights are [1/(28), 21/(28), 6/(28), 0/(28)], summing exactly to one. Compute E[X] exactly as a reduced positive-denominator fraction; do not replace it with the ordinary average of distinct values.

• G001 — The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -25), (1, -25), (2, -4), (3, 1018)}. In the displayed order of S, the nonnegative probability weights are [1/(28), 21/(28), 6/(28), 0/(28)], summing exactly to one. Compute E[X] exactly as a reduced positive-denominator fraction; do not replace it with the ordinary average of distinct values.

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

source roles = [qf-expectation-weighted, no-domain-field, expectation, no-form-field]

derived = [[reduced_fraction_RETAIN1_STRING]]

Expectation ORIGINAL weight total through slot0 := sum(0, 1)

Expectation ORIGINAL weight total through slot0 = 1

Expectation ORIGINAL weight total through slot1 := sum(1, 21)

Expectation ORIGINAL weight total through slot1 = 22

Expectation ORIGINAL weight total through slot2 := sum(22, 6)

Expectation ORIGINAL weight total through slot2 = 28

Expectation ORIGINAL weight total through slot3 := sum(28, 0)

Expectation ORIGINAL weight total through slot3 = 28

source roles = [[0, 1, 2, 3], powerset(S), [1, 21, 6, 0], 28, [-25, -25, -4, 1018], not-supplied]

derived = [[0, 1, 2, 3], 28, Yes]

source roles = [0, 0, expectation, -25, not-supplied, not-supplied, not-supplied, not-supplied]

derived = [-25]

Expectation ORIGINAL slot0 weight times transformed value := mul(1, -25)

Expectation ORIGINAL slot0 weight times transformed value = -25

Expectation ORIGINAL weighted numerator through slot0 := sum(0, -25)

Expectation ORIGINAL weighted numerator through slot0 = -25

source roles = [1, 1, expectation, -25, not-supplied, not-supplied, not-supplied, not-supplied]

derived = [-25]

Expectation ORIGINAL slot1 weight times transformed value := mul(21, -25)

Expectation ORIGINAL slot1 weight times transformed value = -525

Expectation ORIGINAL weighted numerator through slot1 := sum(-25, -525)

Expectation ORIGINAL weighted numerator through slot1 = -550

source roles = [2, 2, expectation, -4, not-supplied, not-supplied, not-supplied, not-supplied]

derived = [-4]

Expectation ORIGINAL slot2 weight times transformed value := mul(6, -4)

Expectation ORIGINAL slot2 weight times transformed value = -24

Expectation ORIGINAL weighted numerator through slot2 := sum(-550, -24)

Expectation ORIGINAL weighted numerator through slot2 = -574

source roles = [3, 3, expectation, 1018, not-supplied, not-supplied, not-supplied, not-supplied]

derived = [1018]

Expectation ORIGINAL slot3 weight times transformed value := mul(0, 1018)

Expectation ORIGINAL slot3 weight times transformed value = 0

Expectation ORIGINAL weighted numerator through slot3 := sum(-574, 0)

Expectation ORIGINAL weighted numerator through slot3 = -574

Expectation ORIGINAL weighted numerator over denominator := div(-574, 28)

Expectation ORIGINAL weighted numerator over denominator = -41/2

source roles = [[0, 1, 2, 3], [1, 21, 6, 0], 28, expectation, [-25, -25, -4, 1018]]

derived = [[-25, -525, -24, 0], -574, -41/2]

Reverse checks

• S020 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S019 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator over denominator with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot3 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Expectation ORIGINAL slot3 weight times transformed value with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S015 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot2 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Expectation ORIGINAL slot2 weight times transformed value with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S012 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Expectation ORIGINAL slot1 weight times transformed value with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S009 · premise_and_result_audit — Recompute Expectation ORIGINAL weighted numerator through slot0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Expectation ORIGINAL slot0 weight times transformed value with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S006 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S005 · premise_and_result_audit — Recompute Expectation ORIGINAL weight total through slot3 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Expectation ORIGINAL weight total through slot2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Expectation ORIGINAL weight total through slot1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Expectation ORIGINAL weight total through slot0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.

Result

-41/2

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.