Page 186 - Probability, Random Variables and Random Processes

RANDOM PROCESSES [CHAP 5

From Eq. (5.73), we can express X, as

where Z, = X, = 0 and Zi (i 2 1) are iid r.v.3 with
P(Zi = +1) = p P(Z,= -l)=q= l-p

i + k
Using Eqs. (5.74) and (5.75), we obtain
R,(n, m) = min(n, m) + [nm - min(n, m)](p - q)2
m+(nm-m)(p-q)2 m<n
Rx(n, m) =
n + (nm - nKp - q)2 n < m
Note that ifp = q = 3, then
Rx(n, m) = min(n, m) n, m > 0

5.12. Consider the random process X(t) of Prob. 5.4; that is,

X(t)=Ycosot t2O
where cu is a constant and Y is a uniform r.v. over (0, 1).
(a) Find E[X(t)].
(b) Find the autocorrelation function R,(t, s) of X(t).
(c) Find the autocovariance function Kx(t, s) of X(t).
(a) From Eqs. (2.46) and (2.91), we have E(Y) = 4 and E(y2) = 4. Thus
E[X(t)] = E(Y cos ot) = E(Y) cos at = 4 cos ot
(b) By Eq. (5.7), we have
R,(t, s) = E[X(t)X(s)] = E(Y2 cos wt cos US)
= E(Y~) cos wt cos US = 3 cos ot cos US
(c) By Eq. @.lo), we have
Kx(t, s) = Rdt, s) - ECX(t)lECX(s)l
= 4 COS Ot COS US - 3 cos ot cos os
= COS Ot COS US

5.13. Consider a discrete-parameter random process X(n) = {X,, n 2 1) where the Xis are iid r.v.'s
with common cdf F,(x), mean p, and variance a2.

(a) Find the joint cdf of X(n).
(b) Find the mean of X(n).
(c) Find the autocorrelation function Rdn, m) of X(n).
(d) Find the autocovariance function Kx(n, m) of X(n).