Question 10721: Qualify a Complete Joint Path Law before Taking Marginals

Original question and givens

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[20, 20, 21], [20, 21, 21], [21, 21, 20]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).

• G001 — Use one common finite probability space specified by the complete distinct equal-horizon path list=[[20, 20, 21], [20, 21, 21], [21, 21, 20]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).

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-discrete-stochastic-process

Step-by-step derivation

original support index=0;path=[20, 20, 21];raw mass=[1, 4]

aligned reduced mass=1/4

original support index=1;path=[20, 21, 21];raw mass=[1, 4]

aligned reduced mass=1/4

original support index=2;path=[21, 21, 20];raw mass=[1, 2]

aligned reduced mass=1/2

whole path-law mass sum := sum(1/4, 1/4, 1/2)

whole path-law mass sum = 1

horizon=3;normalized aligned masses=[1/4, 1/4, 1/2]

all nonnegative=Yes;total=1;qualified joint law=Yes

Reverse checks

• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute whole path-law mass sum with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([[20, 20, 21], [20, 21, 21], [21, 21, 20]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

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.