Consider a digital communication system that transmits the digits 0 and 1 through several stages. At each stage, the probability that 0 is received as 1 is 0.25 and the probability that a 1 is received as a 0 is 0.3. Let Xn = 0 if a 0 entered at the nth stage and Xn = 1 if a 1 entered at the nth stage. Then the sequence {Xn, n = 0, 1, 2, ...} is a Markov chain.
a) Construct the transition probability matrix P.
b) What is the probability that a 0 that is entered at the first stage is received as a 0 by the fifth stage?
c) Obtain the stationary probabilities π0 and π1.
d) Let fii^n be the probability that the first return to state i occurs n transmissions after leaving i. That is, fii^n = P(Xn = i, Xk ≠ i for k = 1, 2, 3, ..., n - 1 | X0 = i). Compute f00^1, f00^2, f00^3, f00^n for n ≥ 3.
e) Calculate the mean recurrence time μ0 = ∑(n=1 to ∞) n f00^n.