Question 1205112: Suppose 𝑋1, 𝑋2, . . . is a discrete time Markov chain with the set of state spaces state space 𝑆 = {1, 2, 3, 4, 5, 6} and the transition probability matrix as follows:
𝑃 =
[
0 0 0.5 0.5 0 0
0 0 1 0 0 0
0 0 0 0 0.5 0.5
0 0 0 0 1 0
0.5 0.5 0 0 0 0
1 0 0 0 0 0
]
Find:
(a) 𝑃(𝑋3000000 = 2 | 𝑋0 = 1)
(b) 𝑃(𝑋3000001 = 2 | 𝑋0 = 1)
(c) 𝑃(𝑋3000002 = 2 | 𝑋0 = 1)
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **1. Identify the Communicating Classes**
* **Class 1: {1, 6}**
* State 1 transitions to state 6 with probability 1.
* State 6 transitions to state 1 with probability 1.
* This forms a closed, irreducible, and recurrent class.
* **Class 2: {2, 3, 4, 5}**
* States within this class transition only to other states within the class.
* This forms another closed, irreducible, and recurrent class.
**2. Long-Term Behavior**
* In the long run, a Markov Chain will tend to spend most of its time in the recurrent classes.
**3. Calculate Probabilities**
* **(a) P(X₃₀₀₀₀₀₀ = 2 | X₀ = 1)**
* Since state 1 belongs to Class 1 and state 2 belongs to Class 2, it's impossible to reach state 2 from state 1 in any number of steps.
* **P(X₃₀₀₀₀₀₀ = 2 | X₀ = 1) = 0**
* **(b) P(X₃₀₀₀₀₀₁ = 2 | X₀ = 1)**
* Even after one step, it's still impossible to reach state 2 from state 1 directly.
* **P(X₃₀₀₀₀₀₁ = 2 | X₀ = 1) = 0**
* **(c) P(X₃₀₀₀₀₀₂ = 2 | X₀ = 1)**
* After two steps:
* State 1 transitions to state 6 with probability 1.
* State 6 transitions to state 1 with probability 1.
* It's still impossible to reach state 2 from state 1 in two steps.
* **P(X₃₀₀₀₀₀₂ = 2 | X₀ = 1) = 0**
**In summary:**
* Due to the structure of the transition probability matrix and the distinct communicating classes, it's impossible to reach state 2 from state 1 in any number of steps. Therefore, all the requested probabilities are 0.
**Key Concepts**
* **Communicating Classes:** A set of states is a communicating class if every state in the set can be reached from every other state in the set.
* **Irreducible Class:** A communicating class is irreducible if it cannot be further divided into smaller communicating classes.
* **Recurrent Class:** A state is recurrent if, starting from that state, the probability of eventually returning to that state is 1.
* **Long-Term Behavior:** In the long run, a Markov Chain will tend to spend most of its time in its recurrent classes.
I hope this explanation is helpful! Let me know if you have any further questions.
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