Answer: P(Xn+1 = j | Xn = i) - depends only on current state.
- A P(Xn = j), the marginal probability of being in state j at time n
- B P(Xn+1 = j | Xn = i) - depends only on current state
- C P(Xn+1 = j | all past), conditioning on the entire observed history
- D P(Xi = i) × P(Xj = j), the product of two unrelated marginal probabilities
Correct answer: B. P(Xn+1 = j | Xn = i) - depends only on current state
Explanation: Markov property (memorylessness): P(Xₙ₊₁=j | Xₙ=i, Xₙ₋₁=iₙ₋₁, …) = P(Xₙ₊₁=j | Xₙ=i) = P(i,j). Only the current state i matters; the full history is irrelevant. The matrix of all P(i,j) is the transition matrix.
Venn diagram of the universal set U with events A and B, showing the intersection (A and B), the parts unique to each event, and the complement region outside both.
Concept context
Chance, events, conditional probability, and Bayes theorem