Question 3: Alternating Random Walk (18 marks) Let the stochastic process {Xn : n=0,1,2,...} be a Markov chain on the integers with Xo = 0. If | Xn is even, the conditional probabilities P(Xn+1 = i +1 X = i) = 3/4 and P(Xn+1 = 1 - 1|X, = i) = 1/4 are given, whereas if X, is odd, the conditional probabilities are P(Xn+1 = i +1|X, = i) = 1/4 and P(Xn+1 = i -1|X, = i) = 3/4. a) Calculate E[X] (5 marks] b) Calculate the variance Var[X,] = E[(Xn - E[X])?). [8 marks) c) For the above Markov chain, use a one-step analysis to calculate the probability to reach state 5 before reaching state 0 when starting in state 3. (5 marks) Question 3: Alternating Random Walk (18 marks) Let the stochastic process {Xn : n=0,1,2,...} be a Markov chain on the integers with Xo = 0. If | Xn is even, the conditional probabilities P(Xn+1 = i +1 X = i) = 3/4 and P(Xn+1 = 1 - 1|X, = i) = 1/4 are given, whereas if X, is odd, the conditional probabilities are P(Xn+1 = i +1|X, = i) = 1/4 and P(Xn+1 = i -1|X, = i) = 3/4. a) Calculate E[X] (5 marks] b) Calculate the variance Var[X,] = E[(Xn - E[X])?). [8 marks) c) For the above Markov chain, use a one-step analysis to calculate the probability to reach state 5 before reaching state 0 when starting in state 3