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224 ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

exists, or, equivalently,

exists.

6.12. Let X(t) be the Wiener process with parameter a2. Let

(a) Find the mean and the variance of Y(t).
(b) Find the autocorrelation function of Y(t).
(a) By assumption 3 of the Wiener process (Sec. 5.7), that is, E[X(t)] = 0, we have

E[Y(t)] = E[[X(a) da] = [E[X(a)] da = 0

Then

By Eq. (5.64), R,(a, fi) = a2 min(a, fl); thus, referring to Fig. 6-3, we obtain

(b) Let t > s 2 0 and write

Then, for t > s 2 0,

Fig. 6-3