# Complete Finite-Horizon Joint Path Laws and Marginals

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[0, 0, 1], [0, 1, 1], [1, 1, 0]] 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?).

View answer and explanation([[0, 0, 1], [0, 1, 1], [1, 1, 0]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0, 1], [0, 1, 1], [1, 1, 0]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0001/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 2. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[1, 1, 2], [1, 2, 2], [2, 2, 1]] 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?).

View answer and explanation([[1, 1, 2], [1, 2, 2], [2, 2, 1]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1, 2], [1, 2, 2], [2, 2, 1]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0002/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 3. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[2, 2, 3], [2, 3, 3], [3, 3, 2]] 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?).

View answer and explanation([[2, 2, 3], [2, 3, 3], [3, 3, 2]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[2, 2, 3], [2, 3, 3], [3, 3, 2]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0003/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 4. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[3, 3, 4], [3, 4, 4], [4, 4, 3]] 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?).

View answer and explanation([[3, 3, 4], [3, 4, 4], [4, 4, 3]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[3, 3, 4], [3, 4, 4], [4, 4, 3]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0004/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 5. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[4, 4, 5], [4, 5, 5], [5, 5, 4]] 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?).

View answer and explanation([[4, 4, 5], [4, 5, 5], [5, 5, 4]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[4, 4, 5], [4, 5, 5], [5, 5, 4]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0005/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 6. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[5, 5, 6], [5, 6, 6], [6, 6, 5]] 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?).

View answer and explanation([[5, 5, 6], [5, 6, 6], [6, 6, 5]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[5, 5, 6], [5, 6, 6], [6, 6, 5]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0006/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 7. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[6, 6, 7], [6, 7, 7], [7, 7, 6]] 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?).

View answer and explanation([[6, 6, 7], [6, 7, 7], [7, 7, 6]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[6, 6, 7], [6, 7, 7], [7, 7, 6]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0007/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 8. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[7, 7, 8], [7, 8, 8], [8, 8, 7]] 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?).

View answer and explanation([[7, 7, 8], [7, 8, 8], [8, 8, 7]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[7, 7, 8], [7, 8, 8], [8, 8, 7]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0008/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 9. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[8, 8, 9], [8, 9, 9], [9, 9, 8]] 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?).

View answer and explanation([[8, 8, 9], [8, 9, 9], [9, 9, 8]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[8, 8, 9], [8, 9, 9], [9, 9, 8]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0009/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 10. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[9, 9, 10], [9, 10, 10], [10, 10, 9]] 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?).

View answer and explanation([[9, 9, 10], [9, 10, 10], [10, 10, 9]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[9, 9, 10], [9, 10, 10], [10, 10, 9]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0010/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 11. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[10, 10, 11], [10, 11, 11], [11, 11, 10]] 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?).

View answer and explanation([[10, 10, 11], [10, 11, 11], [11, 11, 10]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[10, 10, 11], [10, 11, 11], [11, 11, 10]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0011/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 12. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[11, 11, 12], [11, 12, 12], [12, 12, 11]] 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?).

View answer and explanation([[11, 11, 12], [11, 12, 12], [12, 12, 11]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[11, 11, 12], [11, 12, 12], [12, 12, 11]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0012/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 13. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[12, 12, 13], [12, 13, 13], [13, 13, 12]] 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?).

View answer and explanation([[12, 12, 13], [12, 13, 13], [13, 13, 12]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[12, 12, 13], [12, 13, 13], [13, 13, 12]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0013/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 14. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[13, 13, 14], [13, 14, 14], [14, 14, 13]] 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?).

View answer and explanation([[13, 13, 14], [13, 14, 14], [14, 14, 13]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[13, 13, 14], [13, 14, 14], [14, 14, 13]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0014/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 15. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[14, 14, 15], [14, 15, 15], [15, 15, 14]] 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?).

View answer and explanation([[14, 14, 15], [14, 15, 15], [15, 15, 14]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[14, 14, 15], [14, 15, 15], [15, 15, 14]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0015/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 16. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[15, 15, 16], [15, 16, 16], [16, 16, 15]] 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?).

View answer and explanation([[15, 15, 16], [15, 16, 16], [16, 16, 15]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[15, 15, 16], [15, 16, 16], [16, 16, 15]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0016/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 17. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[16, 16, 17], [16, 17, 17], [17, 17, 16]] 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?).

View answer and explanation([[16, 16, 17], [16, 17, 17], [17, 17, 16]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[16, 16, 17], [16, 17, 17], [17, 17, 16]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0017/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 18. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[17, 17, 18], [17, 18, 18], [18, 18, 17]] 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?).

View answer and explanation([[17, 17, 18], [17, 18, 18], [18, 18, 17]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[17, 17, 18], [17, 18, 18], [18, 18, 17]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0018/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 19. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[18, 18, 19], [18, 19, 19], [19, 19, 18]] 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?).

View answer and explanation([[18, 18, 19], [18, 19, 19], [19, 19, 18]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[18, 18, 19], [18, 19, 19], [19, 19, 18]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0019/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 20. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[19, 19, 20], [19, 20, 20], [20, 20, 19]] 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?).

View answer and explanation([[19, 19, 20], [19, 20, 20], [20, 20, 19]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[19, 19, 20], [19, 20, 20], [20, 20, 19]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0020/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 21. Qualify a Complete Joint Path Law before Taking Marginals

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?).

View answer and explanation([[20, 20, 21], [20, 21, 21], [21, 21, 20]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[20, 20, 21], [20, 21, 21], [21, 21, 20]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0021/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 22. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[21, 21, 22], [21, 22, 22], [22, 22, 21]] 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?).

View answer and explanation([[21, 21, 22], [21, 22, 22], [22, 22, 21]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[21, 21, 22], [21, 22, 22], [22, 22, 21]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0022/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 23. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[22, 22, 23], [22, 23, 23], [23, 23, 22]] 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?).

View answer and explanation([[22, 22, 23], [22, 23, 23], [23, 23, 22]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[22, 22, 23], [22, 23, 23], [23, 23, 22]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0023/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 24. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[23, 23, 24], [23, 24, 24], [24, 24, 23]] 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?).

View answer and explanation([[23, 23, 24], [23, 24, 24], [24, 24, 23]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[23, 23, 24], [23, 24, 24], [24, 24, 23]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0024/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 25. Qualify a Complete Joint Path Law before Taking Marginals

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[24, 24, 25], [24, 25, 25], [25, 25, 24]] 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?).

View answer and explanation([[24, 24, 25], [24, 25, 25], [25, 25, 24]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)

Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[24, 24, 25], [24, 25, 25], [25, 25, 24]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0025/)

Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate





## 26. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[0, 0, 1], [0, 1, 1], [1, 1, 0]] 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. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(0, [[0, 1/2], [1, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[0, 1/2], [1, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0026/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 27. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[1, 1, 2], [1, 2, 2], [2, 2, 1]] 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. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(1, [[1, 1/4], [2, 3/4]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[1, 1/4], [2, 3/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0027/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 28. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[2, 2, 3], [2, 3, 3], [3, 3, 2]] 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. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(2, [[2, 1/2], [3, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[2, 1/2], [3, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0028/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 29. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[3, 3, 4], [3, 4, 4], [4, 4, 3]] 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. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(0, [[3, 1/2], [4, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[3, 1/2], [4, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0029/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 30. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[4, 4, 5], [4, 5, 5], [5, 5, 4]] 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. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(1, [[4, 1/4], [5, 3/4]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[4, 1/4], [5, 3/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0030/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 31. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[5, 5, 6], [5, 6, 6], [6, 6, 5]] 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. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(2, [[5, 1/2], [6, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[5, 1/2], [6, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0031/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 32. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[6, 6, 7], [6, 7, 7], [7, 7, 6]] 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. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(0, [[6, 1/2], [7, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[6, 1/2], [7, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0032/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 33. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[7, 7, 8], [7, 8, 8], [8, 8, 7]] 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. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(1, [[7, 1/4], [8, 3/4]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[7, 1/4], [8, 3/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0033/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 34. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[8, 8, 9], [8, 9, 9], [9, 9, 8]] 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. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(2, [[8, 1/2], [9, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[8, 1/2], [9, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0034/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 35. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[9, 9, 10], [9, 10, 10], [10, 10, 9]] 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. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(0, [[9, 1/2], [10, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[9, 1/2], [10, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0035/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 36. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[10, 10, 11], [10, 11, 11], [11, 11, 10]] 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. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(1, [[10, 1/4], [11, 3/4]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[10, 1/4], [11, 3/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0036/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 37. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[11, 11, 12], [11, 12, 12], [12, 12, 11]] 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. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(2, [[11, 1/2], [12, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[11, 1/2], [12, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0037/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 38. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[12, 12, 13], [12, 13, 13], [13, 13, 12]] 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. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(0, [[12, 1/2], [13, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[12, 1/2], [13, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0038/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 39. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[13, 13, 14], [13, 14, 14], [14, 14, 13]] 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. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(1, [[13, 1/4], [14, 3/4]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[13, 1/4], [14, 3/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0039/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 40. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[14, 14, 15], [14, 15, 15], [15, 15, 14]] 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. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(2, [[14, 1/2], [15, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[14, 1/2], [15, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0040/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 41. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[15, 15, 16], [15, 16, 16], [16, 16, 15]] 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. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(0, [[15, 1/2], [16, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[15, 1/2], [16, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0041/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 42. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[16, 16, 17], [16, 17, 17], [17, 17, 16]] 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. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(1, [[16, 1/4], [17, 3/4]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[16, 1/4], [17, 3/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0042/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 43. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[17, 17, 18], [17, 18, 18], [18, 18, 17]] 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. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(2, [[17, 1/2], [18, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[17, 1/2], [18, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0043/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 44. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[18, 18, 19], [18, 19, 19], [19, 19, 18]] 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. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(0, [[18, 1/2], [19, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[18, 1/2], [19, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0044/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 45. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[19, 19, 20], [19, 20, 20], [20, 20, 19]] 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. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(1, [[19, 1/4], [20, 3/4]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[19, 1/4], [20, 3/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0045/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 46. Aggregate a Marginal from the Complete Joint Path Law

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. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(2, [[20, 1/2], [21, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[20, 1/2], [21, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0046/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 47. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[21, 21, 22], [21, 22, 22], [22, 22, 21]] 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. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(0, [[21, 1/2], [22, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[21, 1/2], [22, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0047/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 48. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[22, 22, 23], [22, 23, 23], [23, 23, 22]] 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. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(1, [[22, 1/4], [23, 3/4]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[22, 1/4], [23, 3/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0048/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 49. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[23, 23, 24], [23, 24, 24], [24, 24, 23]] 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. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(2, [[23, 1/2], [24, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[23, 1/2], [24, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0049/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 50. Aggregate a Marginal from the Complete Joint Path Law

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[24, 24, 25], [24, 25, 25], [25, 25, 24]] 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. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.

View answer and explanation(0, [[24, 1/2], [25, 1/2]])

At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[24, 1/2], [25, 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0050/)

Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate





## 51. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[0, 0, 1], [0, 1, 1], [1, 1, 0]] 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. Require a qualified law. time=0, event input=[1]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(0, [1], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [1], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0051/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 52. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[1, 1, 2], [1, 2, 2], [2, 2, 1]] 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. Require a qualified law. time=1, event input=[2]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(1, [2], 3/4)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [2], 3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0052/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 53. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[2, 2, 3], [2, 3, 3], [3, 3, 2]] 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. Require a qualified law. time=2, event input=[3]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(2, [3], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [3], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0053/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 54. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[3, 3, 4], [3, 4, 4], [4, 4, 3]] 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. Require a qualified law. time=0, event input=[4]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(0, [4], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [4], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0054/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 55. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[4, 4, 5], [4, 5, 5], [5, 5, 4]] 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. Require a qualified law. time=1, event input=[5]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(1, [5], 3/4)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [5], 3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0055/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 56. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[5, 5, 6], [5, 6, 6], [6, 6, 5]] 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. Require a qualified law. time=2, event input=[6]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(2, [6], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [6], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0056/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 57. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[6, 6, 7], [6, 7, 7], [7, 7, 6]] 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. Require a qualified law. time=0, event input=[7]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(0, [7], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [7], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0057/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 58. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[7, 7, 8], [7, 8, 8], [8, 8, 7]] 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. Require a qualified law. time=1, event input=[8]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(1, [8], 3/4)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [8], 3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0058/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 59. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[8, 8, 9], [8, 9, 9], [9, 9, 8]] 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. Require a qualified law. time=2, event input=[9]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(2, [9], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [9], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0059/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 60. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[9, 9, 10], [9, 10, 10], [10, 10, 9]] 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. Require a qualified law. time=0, event input=[10]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(0, [10], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [10], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0060/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 61. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[10, 10, 11], [10, 11, 11], [11, 11, 10]] 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. Require a qualified law. time=1, event input=[11]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(1, [11], 3/4)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [11], 3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0061/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 62. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[11, 11, 12], [11, 12, 12], [12, 12, 11]] 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. Require a qualified law. time=2, event input=[12]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(2, [12], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [12], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0062/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 63. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[12, 12, 13], [12, 13, 13], [13, 13, 12]] 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. Require a qualified law. time=0, event input=[13]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(0, [13], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [13], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0063/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 64. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[13, 13, 14], [13, 14, 14], [14, 14, 13]] 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. Require a qualified law. time=1, event input=[14]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(1, [14], 3/4)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [14], 3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0064/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 65. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[14, 14, 15], [14, 15, 15], [15, 15, 14]] 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. Require a qualified law. time=2, event input=[15]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(2, [15], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [15], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0065/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 66. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[15, 15, 16], [15, 16, 16], [16, 16, 15]] 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. Require a qualified law. time=0, event input=[16]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(0, [16], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [16], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0066/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 67. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[16, 16, 17], [16, 17, 17], [17, 17, 16]] 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. Require a qualified law. time=1, event input=[17]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(1, [17], 3/4)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [17], 3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0067/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 68. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[17, 17, 18], [17, 18, 18], [18, 18, 17]] 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. Require a qualified law. time=2, event input=[18]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(2, [18], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [18], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0068/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 69. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[18, 18, 19], [18, 19, 19], [19, 19, 18]] 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. Require a qualified law. time=0, event input=[19]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(0, [19], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [19], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0069/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 70. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[19, 19, 20], [19, 20, 20], [20, 20, 19]] 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. Require a qualified law. time=1, event input=[20]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(1, [20], 3/4)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [20], 3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0070/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 71. One Time-Event Probability with Duplicate-Free Event Semantics

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. Require a qualified law. time=2, event input=[21]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(2, [21], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [21], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0071/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 72. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[21, 21, 22], [21, 22, 22], [22, 22, 21]] 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. Require a qualified law. time=0, event input=[22]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(0, [22], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [22], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0072/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 73. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[22, 22, 23], [22, 23, 23], [23, 23, 22]] 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. Require a qualified law. time=1, event input=[23]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(1, [23], 3/4)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [23], 3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0073/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 74. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[23, 23, 24], [23, 24, 24], [24, 24, 23]] 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. Require a qualified law. time=2, event input=[24]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(2, [24], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [24], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0074/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 75. One Time-Event Probability with Duplicate-Free Event Semantics

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[24, 24, 25], [24, 25, 25], [25, 25, 24]] 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. Require a qualified law. time=0, event input=[25]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.

View answer and explanation(0, [25], 1/2)

Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [25], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0075/)

Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate





## 76. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[0, 0, 1], [0, 1, 1], [1, 1, 0]] 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. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(0, 1, [[[0, 0], 1/4], [[0, 1], 1/4], [[1, 1], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[0, 0], 1/4], [[0, 1], 1/4], [[1, 1], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0076/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 77. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[1, 1, 2], [1, 2, 2], [2, 2, 1]] 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. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(1, 2, [[[1, 2], 1/4], [[2, 1], 1/2], [[2, 2], 1/4]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[1, 2], 1/4], [[2, 1], 1/2], [[2, 2], 1/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0077/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 78. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[2, 2, 3], [2, 3, 3], [3, 3, 2]] 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. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(2, 0, [[[2, 3], 1/2], [[3, 2], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[2, 3], 1/2], [[3, 2], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0078/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 79. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[3, 3, 4], [3, 4, 4], [4, 4, 3]] 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. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(0, 1, [[[3, 3], 1/4], [[3, 4], 1/4], [[4, 4], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[3, 3], 1/4], [[3, 4], 1/4], [[4, 4], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0079/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 80. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[4, 4, 5], [4, 5, 5], [5, 5, 4]] 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. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(1, 2, [[[4, 5], 1/4], [[5, 4], 1/2], [[5, 5], 1/4]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[4, 5], 1/4], [[5, 4], 1/2], [[5, 5], 1/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0080/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 81. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[5, 5, 6], [5, 6, 6], [6, 6, 5]] 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. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(2, 0, [[[5, 6], 1/2], [[6, 5], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[5, 6], 1/2], [[6, 5], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0081/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 82. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[6, 6, 7], [6, 7, 7], [7, 7, 6]] 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. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(0, 1, [[[6, 6], 1/4], [[6, 7], 1/4], [[7, 7], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[6, 6], 1/4], [[6, 7], 1/4], [[7, 7], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0082/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 83. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[7, 7, 8], [7, 8, 8], [8, 8, 7]] 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. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(1, 2, [[[7, 8], 1/4], [[8, 7], 1/2], [[8, 8], 1/4]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[7, 8], 1/4], [[8, 7], 1/2], [[8, 8], 1/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0083/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 84. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[8, 8, 9], [8, 9, 9], [9, 9, 8]] 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. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(2, 0, [[[8, 9], 1/2], [[9, 8], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[8, 9], 1/2], [[9, 8], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0084/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 85. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[9, 9, 10], [9, 10, 10], [10, 10, 9]] 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. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(0, 1, [[[10, 10], 1/2], [[9, 10], 1/4], [[9, 9], 1/4]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[10, 10], 1/2], [[9, 10], 1/4], [[9, 9], 1/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0085/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 86. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[10, 10, 11], [10, 11, 11], [11, 11, 10]] 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. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(1, 2, [[[10, 11], 1/4], [[11, 10], 1/2], [[11, 11], 1/4]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[10, 11], 1/4], [[11, 10], 1/2], [[11, 11], 1/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0086/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 87. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[11, 11, 12], [11, 12, 12], [12, 12, 11]] 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. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(2, 0, [[[11, 12], 1/2], [[12, 11], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[11, 12], 1/2], [[12, 11], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0087/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 88. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[12, 12, 13], [12, 13, 13], [13, 13, 12]] 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. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(0, 1, [[[12, 12], 1/4], [[12, 13], 1/4], [[13, 13], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[12, 12], 1/4], [[12, 13], 1/4], [[13, 13], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0088/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 89. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[13, 13, 14], [13, 14, 14], [14, 14, 13]] 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. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(1, 2, [[[13, 14], 1/4], [[14, 13], 1/2], [[14, 14], 1/4]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[13, 14], 1/4], [[14, 13], 1/2], [[14, 14], 1/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0089/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 90. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[14, 14, 15], [14, 15, 15], [15, 15, 14]] 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. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(2, 0, [[[14, 15], 1/2], [[15, 14], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[14, 15], 1/2], [[15, 14], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0090/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 91. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[15, 15, 16], [15, 16, 16], [16, 16, 15]] 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. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(0, 1, [[[15, 15], 1/4], [[15, 16], 1/4], [[16, 16], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[15, 15], 1/4], [[15, 16], 1/4], [[16, 16], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0091/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 92. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[16, 16, 17], [16, 17, 17], [17, 17, 16]] 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. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(1, 2, [[[16, 17], 1/4], [[17, 16], 1/2], [[17, 17], 1/4]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[16, 17], 1/4], [[17, 16], 1/2], [[17, 17], 1/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0092/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 93. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[17, 17, 18], [17, 18, 18], [18, 18, 17]] 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. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(2, 0, [[[17, 18], 1/2], [[18, 17], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[17, 18], 1/2], [[18, 17], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0093/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 94. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[18, 18, 19], [18, 19, 19], [19, 19, 18]] 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. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(0, 1, [[[18, 18], 1/4], [[18, 19], 1/4], [[19, 19], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[18, 18], 1/4], [[18, 19], 1/4], [[19, 19], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0094/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 95. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[19, 19, 20], [19, 20, 20], [20, 20, 19]] 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. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(1, 2, [[[19, 20], 1/4], [[20, 19], 1/2], [[20, 20], 1/4]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[19, 20], 1/4], [[20, 19], 1/2], [[20, 20], 1/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0095/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 96. A Two-Time Joint Distribution Keeps the Requested Time Order

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. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(2, 0, [[[20, 21], 1/2], [[21, 20], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[20, 21], 1/2], [[21, 20], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0096/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 97. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[21, 21, 22], [21, 22, 22], [22, 22, 21]] 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. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(0, 1, [[[21, 21], 1/4], [[21, 22], 1/4], [[22, 22], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[21, 21], 1/4], [[21, 22], 1/4], [[22, 22], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0097/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 98. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[22, 22, 23], [22, 23, 23], [23, 23, 22]] 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. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(1, 2, [[[22, 23], 1/4], [[23, 22], 1/2], [[23, 23], 1/4]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[22, 23], 1/4], [[23, 22], 1/2], [[23, 23], 1/4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0098/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 99. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[23, 23, 24], [23, 24, 24], [24, 24, 23]] 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. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(2, 0, [[[23, 24], 1/2], [[24, 23], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[23, 24], 1/2], [[24, 23], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0099/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate





## 100. A Two-Time Joint Distribution Keeps the Requested Time Order

Use one common finite probability space specified by the complete distinct equal-horizon path list=[[24, 24, 25], [24, 25, 25], [25, 25, 24]] 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. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.

View answer and explanation(0, 1, [[[24, 24], 1/4], [[24, 25], 1/4], [[25, 25], 1/2]])

Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[24, 24], 1/4], [[24, 25], 1/4], [[25, 25], 1/2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0100/)

Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations/validation-report.json)