Calculator Input
Example Data Table
| From / To | State 1 | State 2 | State 3 | Row Total |
|---|---|---|---|---|
| State 1 | 0.70 | 0.20 | 0.10 | 1.00 |
| State 2 | 0.30 | 0.40 | 0.30 | 1.00 |
| State 3 | 0.20 | 0.30 | 0.50 | 1.00 |
Formula Used
A stationary distribution is a probability vector π that remains unchanged after applying the transition matrix P.
The main equation is:
πP = π
The probabilities must also satisfy:
π1 + π2 + π3 + ... + πn = 1
This calculator uses repeated multiplication. It starts with equal state probabilities. Then it multiplies the vector by the transition matrix until the values stop changing within the selected tolerance.
How to Use This Calculator
- Select the number of Markov states.
- Enter the transition probability for each row and column.
- Make sure every row adds to 1.
- Choose the iteration limit and tolerance.
- Use automatic row normalization only when your inputs are proportional weights.
- Press the calculate button.
- Review the stationary probability for every state.
- Export the result as CSV or PDF when needed.
Understanding Markov Stationary Distribution
What It Means
A Markov stationary distribution shows the long run share of time spent in each state. It is useful when future movement depends only on the current state. The earlier path is not needed. This idea appears in statistics, queue models, finance, weather models, machine learning, and reliability studies.
Why It Matters
Many systems move between repeated conditions. A customer may be active, inactive, or lost. A machine may be working, idle, or failed. A user may visit one page, another page, or leave. The transition matrix records these movements. The stationary result explains the expected balance after many steps.
How Rows Work
Each row describes one current state. Each column describes the next state. Therefore, every row should add to one. A row total below one means missing probability. A row total above one means too much probability. The calculator checks this before solving. It can also normalize rows when values are entered as weights.
Advanced Options
The tolerance controls precision. A smaller tolerance gives a stricter answer. It may require more iterations. The maximum iteration field prevents endless calculation. This is helpful for slow mixing chains. The convergence message tells whether the selected tolerance was reached.
Reading the Output
The result table lists each state. The stationary probability is shown as a decimal. The percent share is also shown. A value of 0.42 means the chain spends about forty two percent of the long run time in that state. This does not predict one exact next step. It describes repeated behavior over many transitions.
Practical Use
Use this tool for checking transition assumptions. Compare different matrices. Test marketing retention states. Study movement between risk classes. Review page navigation patterns. Model equipment states. Export results for reports. Keep rows valid and meaningful. Better input data gives better stationary estimates.
FAQs
What is a stationary distribution?
It is a probability vector that stays unchanged after multiplying by the transition matrix. It describes long run state proportions.
Does every Markov chain have one?
Many finite chains have at least one stationary distribution. A unique stable result usually needs suitable chain structure and reachable states.
Why must each row sum to one?
Each row represents all possible next moves from one current state. Total probability for those moves must equal one.
What does tolerance mean?
Tolerance is the stopping rule. The calculator stops when the largest change between iterations is smaller than this value.
What is automatic normalization?
It converts each row into proportions. Use it when entries are weights, not final probabilities.
Can I use more than three states?
Yes. This page supports two through six states, which covers many practical teaching and reporting examples.
What if the result does not converge?
Increase iterations, check the matrix, or review whether the chain has periodic or disconnected behavior.
Is this result a next-step prediction?
No. It gives long run proportions. It does not guarantee which state will happen on the next transition.