4. Consider a discrete-time Markov chain with the following probability transition matrix 0 0 0 0 7 1-3-y P = 1-I-VVO T 0 0 1 0 Is it possible to choose values for a and y so that the Markov chain has the following properties? In each case, state the values of a and y, or give a brief reason why it is not possible. (a) The Markov chain has period 2. [2) (b) The Markov chain is reducible. (c) The Markov chain has at least one transient state. UNN (d) The Markov chain has invariant distribution (1/4, 1/4, 1/4, 1/4).The fact that the F'F'F usually bows away from the origin implies that ii? getting more of one good means getting more of another good. as the production of any good increases. there is an increase in the opportunity cost of producing it. getting more of one good means getting less of another good- as the production of any good increases. there is a decrease in the opportunity cost of producing it. 2. Recall that Markov's inequality says that if T' is a positive-valued random variable with mean E(T) then E(T) P(T > 1 ) 0 there holds P(1X - #| Ska) 21 - 1 That is. with probability at least 1 - 2. X stays within & standard deviations around its mean