Discrete Markov Chain and Transition Matrix

A discrete Markov chain evolves states through a transition matrix; conditioned on the current state, the next state does not depend on earlier history. The notation π_n+1 = π_nP; P_ij = P(X_n+1=j | X_n=i) shows which quantities are connected. Here, π_n: Row vector of state probabilities at step n (dimensionless, from 0 to 1); P_ij: Transition probability from state i to state j (dimensionless, from 0 to 1); n: Discrete step count (count). Transition and state probabilities are dimensionless; every row of P is nonnegative and sums to one. An application depends on whether the supplied quantities match the definitions and reference directions. Common uses include Model state changes, queues, reliability, random walks and long-run visit frequencies.. Read the variables and their dimensions before substituting numbers; this often reveals whether the result should be a length, rate, probability, or dimensionless statistic.

Start with a diagram or a list of known quantities, assign each one a symbol, and convert units to a consistent form. Then choose the appropriate equation and substitute values only afterward. Example: For P=[[0.8,0.2],[0.3,0.7]] and π₀=(1,0), one step gives π₁=(0.8,0.2). The stationary distribution is π=(0.6,0.4), since πP=(0.6·0.8+0.4·0.3, 0.6·0.2+0.4·0.7)=(0.6,0.4). The entries remain nonnegative and sum to one. Verification is part of the calculation: This page uses row vectors, so the distribution advances as π_n+1=π_nP Also compare the magnitude and unit with the context. Round only at the end, since early rounding can affect multi-step calculations, especially when small differences or sums are involved.

This page uses row vectors, so the distribution advances as π_n+1=π_nP. Each row of P must contain nonnegative entries summing to one. The Markov property says that earlier states add no predictive information about the next step once the current state is known. A stationary distribution satisfies πP=π; uniqueness or convergence requires additional conditions such as irreducibility and aperiodicity. These are not minor footnotes: when assumptions fail, the formula can produce a precise but inappropriate result. Before calculating, identify which values are observed, estimated, or modeled, and whether special cases change the equation. Model state changes, queues, reliability, random walks and long-run visit frequencies. For numerical methods, also check step size and stability; for statistics, sampling design and distribution; for calculus, differentiability, boundary, and orientation. Use another method when these conditions do not hold.

Both relations belong to the same topic, but answer different questions. “Discrete Markov Chain and Transition Matrix”: A discrete Markov chain evolves states through a transition matrix; conditioned on the current state, the next state does not depend on earlier history. The companion relation “Eigenvalues and Eigenvectors” An eigenvector keeps its direction under a linear map; its eigenvalue gives the scale factor. The eigenvalues are found from the characteristic equation. Choose based on the quantity sought and the model: Model state changes, queues, reliability, random walks and long-run visit frequencies. The worked example For P=[[0.8,0.2],[0.3,0.7]] and π₀=(1,0), one step gives π₁=(0.8,0.2). The stationary distribution is π=(0.6,0.4), since πP=(0.6·0.8+0.4·0.3, 0.6·0.2+0.4·0.7)=(0.6,0.4). The entries remain nonnegative and sum to one. shows which quantities actually enter this relation. Compare both formulas using the same data and identify the companion’s extra assumption. This clarifies which relation fits.

With row vectors, advance the distribution by π_n+1=π_nP. In the example, π=(0.6,0.4) is stationary because πP=π. This alone does not show that every starting distribution converges to π. Convergence generally needs further properties such as irreducibility and aperiodicity; periodic chains can oscillate. Before multiplying, check that each row of P sums to one and that π is written as a row vector. This orientation determines whether the transition matrix is multiplied on the left or right. For finite chains, solve πP=π together with the normalization ∑π_i=1.