Question 11338: A Row Initial Law Advances by mu*P

Original question and givens

Use one fixed finite time-homogeneous transition table=[[13/15, 2/15], [7/8, 1/8]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[13/14, 1/14] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

• G001 — Use one fixed finite time-homogeneous transition table=[[13/15, 2/15], [7/8, 1/8]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[13/14, 1/14] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-finite-markov-chain

Step-by-step derivation

transition row0:raw rational pairs=[[13, 15], [2, 15]]

transition row0:normalized=[13/15, 2/15]

transition row0 total := sum(13/15, 2/15)

transition row0 total = 1

transition row0:weights=[13/15, 2/15];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

transition row1:raw rational pairs=[[7, 8], [1, 8]]

transition row1:normalized=[7/8, 1/8]

transition row1 total := sum(7/8, 1/8)

transition row1 total = 1

transition row1:weights=[7/8, 1/8];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

complete transition matrix=[[13/15, 2/15], [7/8, 1/8]];row sums=[1, 1]

nonnegative=Yes;row_stochastic=Yes;rows=current state, columns=next state

initial law:raw rational pairs=[[13, 14], [1, 14]]

initial law:normalized=[13/14, 1/14]

initial law total := sum(13/14, 1/14)

initial law total = 1

initial law:weights=[13/14, 1/14];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

initial row law=[13/14, 1/14];transition=[[13/15, 2/15], [7/8, 1/8]]

next[j]=sum_i initial[i]*P[i][j] over every current state

next state0 current0 term := mul(13/14, 13/15)

next state0 current0 term = 169/210

next state0 current1 term := mul(1/14, 7/8)

next state0 current1 term = 1/16

next state0 total := sum(169/210, 1/16)

next state0 total = 1457/1680

next state1 current0 term := mul(13/14, 2/15)

next state1 current0 term = 13/105

next state1 current1 term := mul(1/14, 1/8)

next state1 current1 term = 1/112

next state1 total := sum(13/105, 1/112)

next state1 total = 223/1680

Reverse checks

• S017 · premise_and_result_audit — Recompute next state1 total with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute next state1 current1 term with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute next state1 current0 term with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute next state0 total with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute next state0 current1 term with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute next state0 current0 term with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute initial law total with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute transition row1 total with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute transition row0 total with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([13/14, 1/14], [[13/15, 2/15], [7/8, 1/8]], [1457/1680, 223/1680])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.