7.45. Let Xn be an irreducible recurrent Markov chain of period d≥ 2. Let P be its transition probability matrix and Pd its d-step transition matrix. Show that the state space is a union of d disjoint sets Eo,... Ed-1 such that (a) Each E is an equivalence class for Pd. (b) If Xo Є Eo, then X₁ = E₁, X2 Є E3,...,Xd-1 € Ed-1 and Xd Є Eo.

Suppose that X0, X1, X2, ... form a Markov chain on the state space {1, 2}. Assume that P(X0 = 1) = P(X0 = 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q11 = 1/2, Q12 = 1/2, Q21 = 1/3, Q22 = 2/3. Find limn→∞ P(Xn = 1).

8. List the Gauss–Markov conditions required for applying a t & F-tests.

For the attached transition probability matrix for a Markov chain. a) Compute P{X2 = 3|X0 = 3}.b) Compute limn→∞ P{Xn = 2|X0 = 4}.(Let x denote this probability, and condition on the first transition when the Markov chain begins in state 4.)

Suppose that X k is a time-homogenous Markov chain. Show thatP{X3= j3, X2= j2|X0= j0,X1 = j1}= P{X3 = j3 | X2 = j2} P{X2 = j2|X1 = j1}.

Consider the problem of sending a binary message, 0 or 1, through a signal channelconsisting of several stages, where transmission through each stage is subject to a fixedprobability of error α. Suppose that X0 = 0 is the signal that is sent and let Xn, be thesignal that is received at the nth stage. Assume that {Xn} is a Markov chain with transitionprobabilities, Poo = P11 = 1- α and P01 = P10 = α, where 0 < α < 1.(a) Determine P {Xo = 0, X1 = 0, X2 = 0}, the probability that no error α occurs up tostage n = 2.(b) Determine the probability that a correct signal is received at stage 2.

Suppose that a production process changes state according to a Markov chain on [25] state space S = {0, 1, 2, 3} whose transition probability matrix is given by a) Determine the limiting distribution for the process. b) Suppose that states 0 and 1 are “in-control,” while states 2 and 3 are deemed “out-of-control.” In the long run, what fraction of time is the process out-of-control?

A cellphone provider classifies its customers as low users (less than 400 minutes per month) or high users (400 or more minutes per month). Studies have shown that 80%  of people who were low users one month will be low users the next month, and that 70% of the people who were high users one month will high users next month.  a. Set up a 2x2 stochastic matrix with columns and rows labeled L and H that displays these transitions   b. Suppose that during the month of January, 50% of the customers are low users. What percent of customers will be low users in February? In March?

For each of the following transition matrices, do the following: (1) Determine whether the Markov chain is irreducible. (2) Find a stationary distribution π; is the stationary distribution unique? (3)Give the period of each state. (4) Without using any software package, find P200 approximately.