Page 190 - Probability, Random Variables and Random Processes

RANDOM PROCESSES [CHAP 5

where p, = E[X(l)].

Then, for any t and s and using Eq. (4.108) and the property of the stationary independent increments, we
have

The only solution to the above functional equation is f(t) = ct, where c is a constant. Since c = f(1) =
E[X(l)], we obtain

5.22. Let {X(t), t 2 0) be a random process with stationary independent increments, and assume that
X(0) = 0. Show that

(4 Var[X(t)] = aI2t (5.1 01)
(4 Var[X(t) - X(s)] = a, 2(t - s) t > s (5.1 02)
where a12 = Var[X(l)].

(a) Let g(t) = Var[X(t)] = Var[X(t) - X(O)]
Then, for any t and s and using Eq. (4.1 12) and the property of the stationary independent increments,
we get

which is the same functional equation as Eq. (5.100). Thus, g(t) = kt, where k is a constant. Since
k = g(1) = Var[X(l)], we obtain

(b) Let t > s. Then

Thus, using Eq. (5.1 Ol), we obtain

5.23. Let {X(t), t 2 0) be a random process with stationary independent increments, and assume that
X(0) = 0. Show that

where aI2 = Var[X(l)].