Probability and Stochastic Processes

For this Markov chain, we have: ao = a1 = = which implies that a2 = P(Xo = 2) = 1-do - 01 = 5. Now, E[X2] = >iP(X2 = 1) i=0 = OP(X2 = 0) + 1P(X2 =1) + 2P(X2 =2) = P(X2 = 1) + 2P(X2 = 2). But, P(X2 = k) = > Pra; where k = 1, 2, a; = P(Xo = i) and PR is the ikth element of P2. 1=0 P2 = NI-WING/- NIHWI-WI- NIH O NIE = For k = 1, we obtain: P(X2 = 1) = Phai 1=0 = aopol+ alPhi + a2P31 = 18 + 1 2 1 =- 9 45 + 30 10+4+3 = 90 17 = 902 For k = 2, we obtain: P(X2 = 2) = Phai 1=0 = + 18 + 7 2 = + + 45 9 15 7 + 10 + 3 = 90 20 = 4 = Hence, E[X2] = P(X2 = 1) + 2P(X2 = 2) 17 = + 2 90 17 8 = + 90 17 + 80 90 97 = 1.0778. 90QUESTION 5 A Markov chain {X, n > 0) with states 0, 1, 2 has the transition probability matrix P = IN|- ON/H If P(Xo = 0) = P(Xo = 1} = =, calculate E[X2].QUESTION 3 A Markov chain {Xn, n > 0) with three states 0, 1, 2 has the transition probability matrix P given by: O P = ON/HAIW WIN If P(Xo = 0) = - and P(Xo = 1} = -, calculate E[X3]