# Transition Matrix for Finite State Random Walk

• Mark53
In summary, a simple random walk on a finite state space involves a random process that either increases or decreases by 1 with certain probabilities, and remains unchanged otherwise. The transition matrix for this process is specified as a matrix with probabilities of increasing, decreasing, and remaining unchanged in each row. For N = 2, the probability of the process being in state 0 after 2 steps can be calculated by finding the probability of being in state 0 after 2 transitions using the given matrix.
Mark53

## Homework Statement

Define a simple random walk Yn on a finite state space S = {0, 1, 2, . . . , N} to be a random process that
• increases by 1, when possible, with probability p,
• decreases by 1, when possible, with probability 1 − p, and
• remains unchanged otherwise.
(a) Specify the transition matrix for Yn.

(b) Assume that N = 2 and initially, the process is evenly distributed across S. Calculate the probability the process is in state 0 after 2 steps.

## The Attempt at a Solution

\begin{pmatrix}
1-p & p & 0 \\ 1-p & 0 & p \\ 0 & 1-p & p

would this matrix be correct not sure about the first entry

b)

Just need to calculate P^2 and see what the probability is in state 0.
Need the correct matrix to do this first

Yes that matrix looks like it implements the rules you wrote.

FactChecker said:
The probabilities of increasing one = p and decreasing one = (1-p) total to 1. So there is no possible such thing as "otherwise".
The 'otherwise' is needed because in some cases one cannot do an increase, and in other cases one cannot decrease. The 'otherwise' pushes the unused probability into the diagonal entry in that row. That tells us what to do in rows 0 and 2.

FactChecker
I stand corrected. I missed the "if possible" and the "Assume N=2". Sorry. I agree that the matrix is correct for N=2.
andrewkirk said:
The 'otherwise' is needed because in some cases one cannot do an increase, and in other cases one cannot decrease. The 'otherwise' pushes the unused probability into the diagonal entry in that row. That tells us what to do in rows 0 and 2.
You are correct. I deleted my wrong post. Sorry.

## What is a transition matrix for finite state random walk?

A transition matrix for finite state random walk is a square matrix that represents the probabilities of transitioning from one state to another in a random walk. It is commonly used in fields such as computer science, mathematics, and physics to model and analyze various processes.

## How is a transition matrix for finite state random walk constructed?

A transition matrix for finite state random walk is constructed by assigning probabilities to each possible transition from one state to another. The probabilities are typically represented as decimal numbers between 0 and 1, with the requirement that the sum of probabilities for each row of the matrix must equal 1.

## What does each element in a transition matrix represent?

Each element in a transition matrix represents the probability of transitioning from one state to another. For example, the element in the first row and first column represents the probability of transitioning from the first state to itself, while the element in the second row and third column represents the probability of transitioning from the second state to the third state.

## How is a transition matrix used in a random walk simulation?

In a random walk simulation, a transition matrix is used to determine the next state in the random walk. The current state is multiplied by the transition matrix, and the resulting vector represents the probabilities of transitioning to each state. A random number is then generated, and based on the probabilities in the vector, the next state is determined.

## What are some applications of a transition matrix for finite state random walk?

A transition matrix for finite state random walk has various applications, such as analyzing the behavior of molecules in a chemical reaction, modeling the movement of particles in a fluid, and simulating the behavior of animals in their natural habitats. It is also used in machine learning algorithms, such as Markov Decision Processes, for decision-making and prediction tasks.

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