General first-order markov chain of 2 states

Overall, understanding both the initial probabilities and transition probabilities is crucial in analyzing a first-order Markov chain. In summary, the conversation discusses the importance of knowing the initial and transition probabilities in a general first-order Markov chain. These probabilities can help determine the probability of each state at any given time and provide additional information about the system's behavior over time.
  • #1
mitra_khanoom
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Hi,
Hope you can give me an answer regarding this trellis diagram. why in this picture of a general first-order markov chain of 2 states,we should know the prob. of each state at each time?

A general first-order markov chain, can be Time-dependent(non stationary) so Transition prob. can change by time. That's why we have E(time) in the picture. but by having the initial Prob. of each state (P(t=1)), and Transition prob. i can calculate prob. of each state in any given time, so The state probabilities at each time (P(t=2,t=3,...)) are Extra informations. am i right or i ignore something?
pic.Statistics in Volcanology (google books) page 167 MArkov chain
 

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Yes, you are correct. Knowing the initial probabilities of each state (P(t=1)), along with the transition probabilities, can help to determine the probability of each state at any given time. The state probabilities at each time (P(t=2,t=3,...)) are additional information that can be used to gain a better understanding of the system and how it behaves over time.
 

Related to General first-order markov chain of 2 states

1. What is a general first-order Markov chain of 2 states?

A general first-order Markov chain of 2 states is a mathematical model used to describe the transition of a system between two distinct states over time. It follows the Markov property, which states that the probability of transitioning to a particular state in the future depends only on the current state and not on any previous states.

2. How does a general first-order Markov chain differ from other types of Markov chains?

A general first-order Markov chain only has two distinct states, while other types of Markov chains may have multiple states. Additionally, the transition probabilities between states in a general first-order Markov chain are fixed and do not change over time.

3. What are some real-world applications of a general first-order Markov chain of 2 states?

General first-order Markov chains are commonly used in finance, economics, and engineering to model the behavior of systems with two distinct outcomes. They can also be applied in fields such as speech recognition and natural language processing.

4. How is the transition matrix calculated in a general first-order Markov chain of 2 states?

The transition matrix in a general first-order Markov chain is calculated by dividing the number of transitions from one state to another by the total number of transitions. This results in a square matrix with the same number of rows and columns as there are states.

5. Can a general first-order Markov chain of 2 states be extended to include more states?

Yes, a general first-order Markov chain of 2 states can be extended to include more states. However, this would require adjusting the transition matrix and may change the dynamics of the system. It is also possible to model more complex systems using higher-order Markov chains with more states.

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