Question

Which of the following is essential for completely specifying a Markov Chain?

• A. Transition probability matrix with state space
• B. Transition probability matrix with initial distribution
• C. Both initial and final distributions
• D. Stochastic process

Hint:

A Markov chain's complete probability law needs both the starting distribution and the one step transition rule, not either alone.

Answer 1

The Correct Option is B

Step 1: A Markov chain is a stochastic process \(\{X_n\}\) on a state space \(S\) satisfying the Markov property \(P(X_{n+1}=j \mid X_n=i, X_{n-1}, \ldots, X_0) = P(X_{n+1}=j \mid X_n=i) = p_{ij}\).

Step 2: The numbers \(p_{ij}\) arranged as a matrix form the transition probability matrix \(P\). This matrix alone tells you how probability moves between states in one step, but it says nothing about which state the chain occupies at time 0.

Step 3: To pin down the full probability law of the chain, that is \(P(X_0=i_0, X_1=i_1, \ldots, X_n=i_n)\) for every path, you also need the initial distribution \(\pi^{(0)}_i = P(X_0=i)\), since \(P(X_0=i_0,\ldots,X_n=i_n) = \pi^{(0)}_{i_0}\, p_{i_0 i_1} p_{i_1 i_2} \cdots p_{i_{n-1} i_n}\).

Step 4: Knowing only the state space or merely calling it a stochastic process gives no numerical probabilities at all, and knowing only initial and final distributions skips the intermediate dynamics. Only the transition probability matrix together with the initial distribution completely fixes the chain.

Answer: option (B).