1. What is a Discrete Time Markov Chain?

A Discrete Time Markov Chain is a stochastic process that undergoes transitions from one state to another on a state space. It must possess the Markov property, meaning the probability of moving to the next state depends only on the current state and not on the sequence of events that preceded it.

2. How Does the Calculator Work?

The calculator uses the fundamental Markov equation:

Where:

• \( P(t) \) — State probability vector at time t
• \( P \) — Transition probability matrix
• \( P(t+1) \) — State probability vector at time t+1

Explanation: The calculator performs iterative matrix multiplication to compute the state probabilities after each time step.

3. Importance of Markov Chains

Details: Markov chains are widely used in statistics, economics, game theory, genetics, and many other fields to model random processes with discrete states and discrete time steps.

4. Using the Calculator

Tips:

• Enter initial state probabilities as comma-separated values (must sum to 1)
• Enter transition matrix with semicolons between rows and commas between values
• Each row of the transition matrix must sum to 1
• Specify the number of steps to calculate

5. Frequently Asked Questions (FAQ)

Q1: What is the Markov property?
A: The Markov property states that the future state depends only on the current state, not on the sequence of events that preceded it.

Q2: What's the difference between discrete and continuous time Markov chains?
A: Discrete time chains evolve in fixed time steps, while continuous time chains can transition at any time.

Q3: What is a transition matrix?
A: A square matrix where each entry (i,j) represents the probability of moving from state i to state j.

Q4: What is a steady state distribution?
A: A state vector that remains unchanged after multiplication by the transition matrix (if it exists).

Q5: Can I use this for absorbing Markov chains?
A: Yes, but you may need many steps to see convergence to absorbing states.