You are given a transition matrix P . Find the steady-state distribution vector: P=[ 1 / 3

step 1 For this problem, we are given the steady -state distribution vector p equals one -third, two -t

You are given a transition matrix P . Find the steady-state...

Key Concepts

Markov Chain

A Markov chain is a mathematical system that undergoes transitions from one state to another on a state space in a memoryless process, where the probability of each future state depends only on the current state and not on the sequence of events that preceded it.

Transition Matrix

A transition matrix is a square matrix that describes the probabilities of transitioning from one state to another in a Markov chain, with each row summing to one. It encapsulates the entire dynamics of the system's state changes.

Steady-State (Stationary) Distribution

The steady-state or stationary distribution is a probability vector that remains invariant after the application of the transition matrix. It satisfies the condition that, when multiplied by the transition matrix, the result is the same vector, representing the long-term behavior of the Markov chain.

Eigenvalue Method

The eigenvalue method is used to find the steady-state distribution by identifying the eigenvector corresponding to the eigenvalue of one. This approach leverages the fact that the steady-state distribution is an eigenvector of the transition matrix associated with the eigenvalue one.

Balance Equations

Balance equations are a set of linear equations that arise from equating the inflow and outflow probabilities for each state in the Markov chain. Solving these equations helps determine the steady-state distribution by ensuring that probability is conserved across transitions.

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