Page 205 - Probability, Random Variables and Random Processes

CHAP. 51 RANDOM PROCESSES

5.46. Consider a Markov chain with two states and transition probability matrix

(a) Find the stationary distribution fi of the chain.
(b) Find limn,, Pn.

(a) By definition (5.52),
pP = p

which yields p, = p,. Since p, + p, = 1, we obtain
P = c3 41

(b) NOW pn =

and lim,, , does not exist.
Pn

5.47. Consider a Markov chain with two states and transition probability matrix

(a) Find the stationary distribution fi of the chain.
(b) Find limn,, Pn.
(c) Find limn,, Pn by first evaluating Pn.
(a) By definition (5.52); we have
pP = p

or

which yields
PI f 3 ~ = PI
2
$PI + 4 ~ = P2
2
Each of these equations is equivalent to p, = 2p2. Since p, + p, = 1, we obtain

(b) Since the Markov chain is regular, by Eq. (5.53), we obtain