Study transition matrices with confidence. Estimate long run probabilities, compare states, and track convergence. Build better statistical insight easily.
| From / To | Sunny | Cloudy | Rainy |
|---|---|---|---|
| Sunny | 0.70 | 0.20 | 0.10 |
| Cloudy | 0.30 | 0.40 | 0.30 |
| Rainy | 0.20 | 0.30 | 0.50 |
This sample matrix models daily weather movement. Long run probabilities show the stable share of time in each weather state.
Steady state condition: πP = π
Total probability condition: π₁ + π₂ + π₃ = 1
Here, P is the transition matrix and π is the steady state vector. The calculator solves the linear system formed by these equations. It also multiplies the initial distribution by P repeatedly to show how probabilities move toward the long run pattern.
A Markov chain describes movement between states. Each move depends only on the current state. It does not depend on the full past. This calculator finds the long run probability of being in each state. That long run vector is called the steady state distribution.
Steady state probabilities help you understand stable behavior. They show what happens after many transitions. This is useful in statistics, finance, operations, weather modeling, and customer behavior analysis. A short term distribution may change fast. The long run pattern is often more informative.
The calculator uses the transition matrix you enter. Each row must sum to one. The program solves the equation πP = π. It also uses the rule that all steady state probabilities add to one. That gives a solvable linear system for three states.
You can enter an initial distribution and a step count. The calculator multiplies that vector by the transition matrix again and again. This creates a path of probability distributions. The table and graph show whether the system moves toward a stable pattern. Many regular chains do converge this way.
If a steady state value is 0.55, that means the process spends about 55 percent of the long run time in that state. It does not mean 55 percent on every single step. It describes long run average behavior. That is why steady state analysis is powerful.
Always verify that every transition probability is valid. No entry should be negative. No entry should exceed one. Each row total must equal one. If your matrix fails these rules, the model is not a proper transition matrix. The calculator warns you when that happens.
Markov chains appear in queue analysis, reliability studies, web navigation, machine learning, and survival modeling. They are also useful for brand switching and credit rating transitions. A steady state calculator saves time and reduces manual algebra. It also makes patterns easier to explain with clear visuals.
It is the long run probability of being in each state of a Markov chain. It shows the stable pattern after many transitions.
Each row lists all possible next state probabilities from one current state. Since one of those outcomes must happen, the row total must equal one.
Not always in a useful form. Some chains do not converge to one stable distribution. Regular and irreducible chains are more likely to have a practical steady state.
It sets the starting probabilities before transitions begin. It affects early steps, but in many chains the long run result becomes independent of that start.
The steady state is the target long run vector. The step results show the path toward that target based on your chosen starting distribution.
Yes. It works well for customer switching, credit ratings, machine states, website movement, and other systems with probabilities between states.
The calculator shows an error. You should fix row totals and ensure every transition value stays between zero and one.
The graph makes convergence easier to see. It helps you compare how each state probability changes over time and where it stabilizes.