Online Tool Store
Markov Chain Calculator
Enter a 2- or 3-state transition matrix to calculate Pⁿ, the projected distribution after n steps, and the stationary distribution with row checks.
🔒 This tool runs entirely in your browser. Your files are never uploaded to a server.
Markov Chain Transition Matrix (P)
Rows must sum to 1.0 (Stochastic Transition Matrix).
Stationary Steady-State Distribution (π)
Long-term equilibrium probability (πP = π)
State Distribution After 5 Steps (πₙ = π₀Pⁿ)
Projected distribution from initial state
Computed n-Step Matrix (Pⁿ)
How it works
- Select the number of states (2 or 3) or load a real-world preset (Weather forecasting or Brand market share loyalty).
- Enter the state-to-state transition probabilities into matrix P, ensuring each row sums to 1.00.
- Specify the initial starting probability vector (π₀) and the desired number of time steps (n).
- The calculator computes the n-step matrix (Pⁿ), the projected future distribution (πₙ), and the long-term stationary steady-state distribution (π).
The formula
n-Step State Vector: πₙ = π₀ × Pⁿ
Stationary Distribution: π × P = π (where ∑ πᵢ = 1)
FAQ
What is a Markov Chain and the Markov Property?
A Markov chain is a stochastic model describing a sequence of possible events in which the probability of transitioning to the next state depends solely on the current state, and not on the sequence of events that preceded it (memorylessness).
What is a stationary steady-state distribution (π)?
The stationary distribution π is a probability vector that remains invariant when multiplied by the transition matrix P (satisfying πP = π). It represents the long-term equilibrium percentage of time the system will spend in each state.
How is the n-step transition probability matrix calculated?
The n-step transition matrix is obtained by raising the one-step transition matrix P to the nth power (Pⁿ) using matrix multiplication.
What is a stochastic matrix?
A right stochastic matrix is a square matrix whose entries are non-negative real numbers and each row sums to exactly 1.0 (representing the total probability across all potential destination states).
How we compare
| Feature | Online Tool Store | Python / NumPy scripts | Manual matrix algebra |
|---|---|---|---|
| Instant in-browser matrix power calculation | ✓ | ✓ | ✗ |
| Real-time row sum validation | ✓ | ✗ | ✗ |
| Ready-to-use real world presets (Weather, Market Share) | ✓ | ✗ | ✗ |