Page 186 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 5) RANDOM PROCESSES

(a) Since the X,'s are iid r.v.'s with common cdf FX(x), the joint cdf of X(n) is given by

(b) The mean of X(n) is
px(n) = E(Xn) = p for all n
(c) If n # m, by Eqs. (5.7) and (5.90),
Rx(n, m) = E(Xn X,) = E(X,)E(X,) = p2
If n = m, then by Eq. (2.31),

Hence,

(4 BY Eq. (5.m

CLASSIFICATION OF RANDOM PROCESSES
5.14. Show that a random process which is stationary to order n is also stationary to all orders lower
than n.

Assume that Eq. (5.14) holds for some particular n; that is,

for any z. Letting x, -, a, we have [see Eq. (3.63)]

and the process is stationary to order n - 1. Continuing the same procedure, we see that the process is
stationary to all orders lower than n.

5.15. Show that if {X(t), t E T) is a strict-sense stationary random process, then it is also WSS.
Since X(t) is strict-sense stationary, the first- and second-order distributions are invariant through time
translation for all T E T. Then we have
px(t) = E[X(t)] = E[X(t + T)] = px(t + t)
and hence the mean function pdt) must be constant; that is,
E[X(t)] = p (constant)
Similarly, we have
E[X(s)X(t)] = E[X(s + z)X(t + T)]
so that the autocorrelation function would depend on the time points s and t only through the difference
( t - s 1. Thus, X(t) is WSS.

5.16. Let (x,, 2 0) be a sequence of iid r.v.'s with mean 0 and variance 1. Show that (X,, n 2 O} is
n
a WSS process.
By Eq. (5.90),
E(Xn) = 0 (constant) for all n