Analyze steady behavior across states using transition probabilities. Test matrices, compare outcomes, and examine convergence. Download reports and charts for faster statistical interpretation workflows.
Enter the number of states, labels, initial distribution, and transition probabilities. Each row must sum to 1.
This example models three weather states and their transition behavior.
| From \ To | Sunny | Cloudy | Rainy | Row Sum |
|---|---|---|---|---|
| Sunny | 0.70 | 0.20 | 0.10 | 1.00 |
| Cloudy | 0.30 | 0.40 | 0.30 | 1.00 |
| Rainy | 0.20 | 0.30 | 0.50 | 1.00 |
Initial distribution: [0.50, 0.30, 0.20]
Use the example button above to load this matrix into the calculator instantly.
A Markov chain transition matrix is written as P. The long run distribution is the stationary vector
π, where:
πP = π and Σπi = 1
The calculator solves that stationary system and also tests convergence through repeated multiplication. Starting from an initial distribution
x0, the step distribution becomes:
xn = x0Pn
When repeated transitions stabilize, the chain approaches the same long run probability pattern. The recurrence estimate for a state uses
1 / πi.
It shows the proportion of time the chain spends in each state after many transitions. It summarizes the chain’s steady behavior rather than one short sequence.
Each row lists the probabilities of moving from one state to every possible next state. Those outcomes must cover all possibilities, so the total probability equals 1.
It is a probability vector that remains unchanged after multiplication by the transition matrix. In symbols, it satisfies πP = π.
Not always. Some chains are periodic or reducible. In those cases, steady convergence can fail or depend on the starting distribution.
It shows the distribution after a chosen number of transitions. This helps compare short term behavior against the estimated long run distribution.
Tolerance defines how close two consecutive distributions must be. Maximum iterations prevents endless looping when stabilization is slow or absent.
For a positive stationary probability, it estimates the average number of steps needed to return to that state. It is computed as 1 / πi.
Yes. This page supports between two and six states. Larger state spaces are possible, but they usually need a bigger custom interface.
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.