Page 203 - Probability, Random Variables and Random Processes

CHAP. 51 RANDOM PROCESSES

Substituting k = 1 and N = 3 in Eq. (5.134), and noting that p + q = 1, we have

Now from Eq. (5.1 28), we have

5.45. Consider the simple random walk X(n) with state space E = (0, 1, 2, . . . , N), where 0 and N are
absorbing states (Prob. 5.38). Let r.v. T, denote the time (or number of steps) to absorption of
X(n) when X, = k, k = 0, 1, . . . , N. Find E(T,).
Let Y(k) = E(G). Clearly, if k = 0 or k = N, then absorption is immediate, and we have

Y(0) = Y(N) = 0
Let the probability that absorption takes m steps when X, = k be defined by
P(k, m) = P(T, = m) m = 1, 2, . . .
Then, we have (Fig. 5-12)

a, Q) Q)
and Y(k) = E(T,) = 2 mP(k, m) = p x mP(k + 1, m - 1) + q C mP(k - 1, m - 1)
m= 1 m= 1 m= 1
Setting m - 1 = i, we get
al

0 1 2 3 k n
Fig. 5-12 Simple random walk with absorbing barriers.