3. (a) If P is the transition matrix of an irreducible Markov chain with finitely many states, show that Q := (1 + P)/2, where I is the identity matrix, is also a transition matrix corresponding to an irreducible Markov chain. (b) Show that P and Q have the same stationary distribution. 4. (a) Consider the Markov process with transition matrix P given by 1/2 1/3 1/6 P = 3/4 0 1/4 0 1 0 Show that the process is irreducible and aperiodic. (b) Calculate the stationary probability distribution for the process in (a)