Markov chain Irreducibility

Consider a Markov chain {Xn : n = 0, 1, 2, ...} with state space {1, 2, 3} and one-step transition probability matrix O NIH NIH P = O 0 O (a) Mark O or X: ( ) The Markov chain is irreducible. ( ) The Markov chain is aperiodic. ( ) The Markov chain is transient. ( ) The Markov chain is recurrent. ( ) The Markov chain is null recurrent. ( ) The Markov chain is ergodic. (b) Calculate P(X5 = 1/X2 = 1). (c) Find limn + P(Xn = 1/X2 = 1).Consider a Markov chain { Xn } with state space S' = {0, 1, 2} and transition matrix P = Let the mapping f : S -> S satisfy f(0) = 0 and f(2) = 1 and assume that f(1) # f(2). If Ym = f(X ), then when is { Y } a Markov chain? Is {Ym } always a Markov chain? In other words, are functions of Markov chains always Markov chains?1. Consider the Markov chain with three states S=[1,2,3), that has the following transition probability matric 0 NI-WIHNI O Draw the state transition diagram for this chain. Write the transition probability matrix for three steps. 2. Write Markov transition matrix for following state transition diagram: Classify of states of the Markov chain. 3. Find eigenvalues and eigenvectors of the transition probability matrix for following system: 0.4 0.6 0.8 rain no rain4 There are given two boxes with 1 white and 2 black balls in each of them. Then arbitrary is taken one ball from 1st box and one ball from 2nd box and put in opposite box. This action is performed many times. The state of Markov chain is determined by number of white balls in the first box. 1) Write the distribution vector of initial state and transition matrix 2) Find the probability that after 3 steps there will be 2 white balls in the first box