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

You are given a transition matrix $P$. Find the steady-state distribution vector. [HINT: See Example $4 .$
$$
P=\left[\begin{array}{ll}
\frac{1}{3} & \frac{2}{3} \\
\frac{1}{2} & \frac{1}{2}
\end{array}\right]
$$

Key Concepts

Markov Chain

A Markov chain is a stochastic process that undergoes transitions from one state to another on a state space. It is characterized by the property that the next state depends only on the current state and not on the sequence of events that preceded it. This property is known as memorylessness or the Markov property.

Transition Matrix

The transition matrix is a square matrix that describes the probabilities of moving from one state to any other state in a single time step. Each element of the matrix represents the probability of transitioning from one state to another, and the rows of the matrix typically sum to one, reflecting the total probability of transitions from a given state.

Steady-State Distribution

The steady-state distribution, also known as the equilibrium distribution, is a probability vector that remains unchanged when the transition matrix is applied. In other words, if the system reaches this distribution, it will stay in that distribution indefinitely. This concept is crucial for understanding the long-term behavior of Markov chains.

Eigenvalues and Eigenvectors

In the context of finding the steady-state distribution, the steady-state vector is the eigenvector associated with the eigenvalue of 1 for the transition matrix. By solving the equation P? = ?, where P is the transition matrix and ? is the steady-state distribution vector, one is essentially finding the eigenvector corresponding to eigenvalue 1.

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