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 $4 .$
$$
P=\left[\begin{array}{ccc}
\frac{1}{2} & \frac{1}{2} & 0 \\
\frac{1}{2} & \frac{1}{2} & 0 \\
\frac{1}{2} & 0 & \frac{1}{2}
\end{array}\right]
$$

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Key Concepts

Markov Chains

A Markov chain is a stochastic process that describes a system transitioning between different states in a probabilistic manner, where the probability of moving to the next state depends solely on the current state. This property is known as the Markov property.

Transition Matrix

A transition matrix is a square matrix that represents the probabilities of moving from each state to every other state in a Markov chain. Each row in the matrix corresponds to a current state and each column to a subsequent state, with all row entries summing to one.

Steady-State Distribution

The steady-state distribution, or stationary distribution, is a probability vector that remains invariant under the application of the transition matrix. It satisfies the equation ? = ?P, meaning that once the Markov chain reaches this distribution, it stays there.

Solving Linear Equations

Determining the steady-state distribution typically involves solving a system of linear equations derived from the steady-state condition ? = ?P along with the constraint that the sum of the probabilities equals one. This method finds the equilibrium point of the Markov process.

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