Let X 0 , X 1 , be a Markov chain with state space S such that i j is the value that X j takes in the j t h state One of the properties that it satisfies is the Markov property P ( X n i n X n 1 i n 1 , , X 0 i 0 ) P ( X n i n X n 1 i n 1 ) , for all i 0 , i 1 , , i n S , n Z 0 Use the Markov property and the total probability theorem to prove the following a) P ( X 3 i 3 X 2 i 2 , X 1 i 1 ) P ( X 3 i 3 X 2 i 2 ) , for all i 1 , i 2 , i 3 S Note This is not exactly the Markov property because it does not condition on X 0 b) P ( X 3 i 3 X 1 i 1 , X 0 i 0 ) P ( X 3 i 3 X 1 i 1 ) , for all i 0 , i 1 , i 3 S c) P ( X 1 i 1 X 2 i 2 , X 3 i 3 ) P ( X 1 i 1 X 2 i 2 ) , for all i 1 , i 2 , i 3 S

Question

Let X 0 , X 1 ,     be a Markov chain with state space S such that i j is the value that X j takes in the j t h state  One of the properties that it satisfies is the Markov property  P ( X n   i n X n 1   i n 1 ,     , X 0   i 0 )   P ( X n   i n X n 1   i n 1 ) , for all i 0 , i 1 ,     , i n S , n Z   0 Use the Markov property and the total probability theorem to prove the following  a) P ( X 3   i 3 X 2   i 2 , X 1   i 1 )   P ( X 3   i 3 X 2   i 2 ) , for all i 1 , i 2 , i 3 S Note  This is not exactly the Markov property because it does not condition on X 0 b) P ( X 3   i 3 X 1   i 1 , X 0   i 0 )   P ( X 3   i 3 X 1   i 1 ) , for all i 0 , i 1 , i 3 S c) P ( X 1   i 1 X 2   i 2 , X 3   i 3 )   P ( X 1   i 1 X 2   i 2 ) , for all i 1 , i 2 , i 3 S

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Question

Let X 0 , X 1 , ... be a Markov chain with state space S such that i j is the value that X

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