Let X[n] be a Markov chain on n ≥ 0 taking values 1 and 2 with one-step transition probabilities, Pij=P{X[n]= j|X[n−1]=i}, 1 ≤ i, j ≤ 2, given in matrix form as We describe the state probabilities at time n by the vector p[n] = [P{X[n] = 1}, P{X[n] = 2}] . (1) Show that p[n] = p[0]P n (2) Draw a two-state transition diagram and label the branches with the one-step transition probabilities . Don’t forget the or self-transitions. (See Figure 8.5-1 for state-transition diagram of a Markov chain.) (3) Given that X[0] = 1, find the probability that the first transition to state 2 occurs at time n.