Understanding Markov Chains: Transition Matrix and State Space Explained

[Poirot1]

What is the transition matrix and state space corresponding to a simple random random walk with absorbing barriers at 1 and 5? I know an absorbing barrier will correspong to a row of zeroes but I don't know what a simple random walk is.Thanks

 

[CaptainBlack]

What is the transition matrix and state space corresponding to a simple random random walk with absorbing barriers at 1 and 5? I know an absorbing barrier will correspong to a row of zeroes but I don't know what a simple random walk is.Thanks

Equal probability of +1, -1.

CB

 

[Poirot1]

Sorry I don't understand what you mean. Can you give me the matrix?

 

[CaptainBlack]

Sorry I don't understand what you mean. Can you give me the matrix?

Something like:

\[A=\left[ \begin{array}{ccccc}1& 0 & 0 & 0 & 0 \\ 0.5 & 0 & 0.5 & 0 & 0 \\ 0 & 0.5 & 0 & 0.5 & 0
\\ 0 & 0 & 0.5 & 0 & 0.5 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right] \]

CB

 

[blue_raver22]

A simple random walk is a type of stochastic process where a random variable moves from one state to another in discrete time steps, with each step being independent of the previous step. In this context, the state space refers to the set of all possible states that the random variable can take, while the transition matrix represents the probabilities of moving from one state to another.

In the case of a simple random walk with absorbing barriers at 1 and 5, the state space would consist of the integers 1 through 5. The transition matrix would be a 5x5 matrix, with rows and columns corresponding to each state. Since the barriers at 1 and 5 are absorbing, the first and last rows of the matrix would be filled with zeroes, indicating that once the random variable reaches these states, it cannot move any further.

To further illustrate, let's say the random variable is currently at state 3. The third row of the transition matrix would indicate the probabilities of moving from state 3 to states 2, 3, 4, and 5. Since this is a simple random walk, each of these probabilities would be equal (0.25). However, the third row would also have a 0 in the first and last columns, representing the fact that the random variable cannot move to states 1 or 5, as they are absorbing barriers.

Overall, the transition matrix and state space are important concepts in understanding Markov chains, as they allow us to model and analyze the behavior of stochastic processes.