Compute finite-state Markov transitions, matrix powers, steady behavior, and forecasts. Check inputs and scenarios carefully. Make better long-run probability decisions with clear statistical insight.
| State | To S1 | To S2 | To S3 | Initial Probability |
|---|---|---|---|---|
| S1 | 0.70 | 0.20 | 0.10 | 1.00 |
| S2 | 0.15 | 0.65 | 0.20 | 0.00 |
| S3 | 0.10 | 0.25 | 0.65 | 0.00 |
This example starts fully in S1. Every row sums to 1. The calculator can estimate future state probabilities and long-run behavior.
An infinite Markov chain uses a transition matrix P. Each row shows the next-step probabilities from one state to all states.
The n-step transition matrix is Pn. This shows the probability of moving between states after n transitions.
If the initial distribution is v0, then the future distribution is vn = v0Pn.
The stationary distribution π satisfies πP = π and Σπi = 1.
Expected return time for state i is 1 / πi, when πi is positive.
An infinite Markov chain studies repeated movement across states. It tracks how a system changes over many steps. The next state depends only on the current state. This makes the model clear and useful. It also makes forecasting easier.
Statisticians use Markov chains in customer retention, weather shifts, queue systems, machine learning, genetics, and finance. A transition matrix captures the chance of moving from one state to another. The matrix becomes the engine of the model. It turns many state questions into direct calculations.
This calculator helps you evaluate n-step probabilities, matrix powers, long-run averages, and steady-state behavior. It also checks whether the chain looks irreducible, regular, or absorbing. Those diagnostics matter. They shape how reliable long-run probability results may be.
The initial distribution describes where the process starts. The matrix power Pn shows transition behavior after many steps. Multiplying the initial distribution by Pn gives the future probability across all states. This output is useful for scenario planning, demand models, and movement analysis.
The stationary distribution is a key result. It represents a probability pattern that remains unchanged after another transition. In practical terms, it estimates the long-run share of time spent in each state. Expected return time adds more depth. It estimates how long it takes, on average, to revisit a state.
Good inputs produce better decisions. Each row must sum to one. Every entry must stay between zero and one. If these rules fail, the model is not a valid stochastic matrix. Clean matrix design improves trust in the output.
You can test several scenarios by changing the matrix or initial distribution. This makes the tool useful for policy tests, churn reduction plans, and risk modeling. Small probability changes can shift long-run outcomes. That is why Markov chain analysis remains valuable in modern statistics.
It is a stochastic process that can continue for unlimited steps. The next move depends only on the present state, not the full past history.
Each row represents all possible next moves from one state. Since one of those moves must occur, the total probability must equal 1.
Pn is the transition matrix after n steps. It shows how likely the process is to move between states across multiple transitions.
It is a probability vector that stays unchanged after another transition. It often describes long-run state behavior when the chain is stable enough.
An absorbing state traps the process once entered. Its self-transition probability equals 1, and all other outgoing probabilities are 0.
It means every state can eventually reach every other state. This property often supports stronger long-run interpretation.
Return time estimates how many steps, on average, it takes to revisit a state. It helps compare state persistence and recurrence.
Export results when you need reporting, auditing, or team sharing. CSV works well for spreadsheets, while PDF is useful for clean presentation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.