Page 201 - Schaum's Outlines - Probability, Random Variables And Random Processes

RANDOM PROCESSES [CHAP 5

walk X(n) with absorbing barriers at states 0 and N. Now if 0 < k < N, then
P(k)=pP(k+ l)+qP(k- 1) k = 1, 2, ..., N - 1
where pP(k + 1) is the probability that A wins the first round and subsequently loses all his money and
qP(k - 1) is the probability that A loses the first round and subsequently loses all his money. Rewriting Eq.
(5.130), we have

which is a second-order homogeneous linear constant-coefficient difference equation. Next, we have
P(0) = 1 and P(N) = 0 (5.1 32)
since if k = 0, absorption at 0 is a sure event, and if k = N, absorption at N has occurred and absorption at
0 is impossible. Thus, finding P(k) reduces to solving Eq. (5.131) subject to the boundary conditions given
by Eq. (5.132). Let P(k) = r". Then Eq. (5.131) becomes

Setting k = 1 (and noting that p + q = I), we get

from which we get r = 1 and r = q/p. Thus,

where c, and c, are arbitrary constants. Now, by Eq. (5.132),

Solving for c, and c, , we obtain

Hence

Note that if N 9 k,

Setting r = q/p in Eq. (5.134), we have

Thus, whenp = q = 3,

5.44. Show that Eq. (5.1 34) is consistent with Eq. (5.1 28).