Page 194 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 51 RANDOM PROCESSES

,
-
x,, = 0 I -a x,, 0
=

Fig. 5-9 Binary communication network.
denotes the digit leaving the nth stage of the channel and X, denotes the digit entering the first
stage. The transition probability matrix of this communication network is often called the
channel matrix and is given by Eq. (5.1 16); that is,

Assume that a = 0.1 and b = 0.2, and the initial distribution is P(X, = 0) = P(X, = 1) = 0.5.
(a) Find the distribution of X, .

(b) Find the distribution of X, when n -, co.
(a) The channel matrix of the communication network is

and the initial distribution is

By Eq. (5.39), the distribution of X, is given by

Letting a = 0.1 and b = 0.2 in Eq. (5.1 17), we get

Thus, the distribution of Xn is