markov transition matrix: describe the qualitative behavior of this process over time; that is, mention no numerical probabilities is π = [1/3, 1/3, 1/3]
From the given information;
the transition probability matrix (TPM) for state 0, 1, 2 is:
The three-step TPM is done by a simple matrix multiplication which is computed as follows:
The steady-state distribution of the Markov Chain is determined by first solving the system πp = π together with the normalizing condition
here;
π₁ + π₂ + π₃ = 1
So;
0.5π₂ + 0.5π₃ = π₁
⇒ π₂ + π₃ = 2π₁
0.5π₁ + 0.5π₃ = π₂
⇒ π₁ + π₃ = 2π₂
0.5π₁ + 0.5π₂ = π₃
⇒ π₁ + π₂ = 2π₃
Thus, π₁ + π₂ + π₃ = 1
This can be re-written as:
1 - π₁ = 2π₁
1 - π₂ = 2π₂
1 - π₃ = 2π₃
∴1 = 3π₁
π₁ = 1/3
1 = 3π₂
π₂ = 1/3
1 = 3π₃
π₃ = 1/3
Hence, the steady-state probability distribution is π = [1/3, 1/3, 1/3]
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