Page 170 - Probability, Random Variables and Random Processes

162 RANDOM PROCESSES [CHAP 5

F,(xl; t,) is known as the first-order distribution of X(t). Similarly, given t, and t,, X(t,) = X, and
X(t2) = X, represent two r.v.3. Their joint distribution is known as the second-order distribution of
X(t) and is given by

In general, we define the nth-order distribution of X(t) by
FX(x1, . . . , x,; ti, . . . , t,) = P{X(tl) I xl, . . . , X(t,) 2 x,)
If X(t) is a discrete-time process, then X(t) is specified by a collection of pmf s:
px(xl, . . . , Xn ; tl, . . . , t,) = P{X(tl) = XI, . . . , X(t,) = x,} (5.4)
If X(t) is a continuous-time process, then X(t) is specified by a collection of pdf s:

The complete characterization of X(t) requires knowledge of all the distributions as n + co. Fortu-
nately, often much less is sufficient.

B. Mean, Correlation, and Covariance Functions:
As in the case of r.v.3, random processes are often described by using statistical averages.
The mean of X(t) is defined by

where X(t) is treated as a random variable for a fixed value of t. In general, p,(t) is a function of time,
and it is often called the ensemble average of X(t). A measure of dependence among the r.v.'s of X(t) is
provided by its autocorrelation function, defined by
Rx(t, s) = ECX(t)X(s)l (5.7)
Note that
RX@, s) = Rx(s, t) (5.8)
and Rx(~, t) = ECX2(t)l (5.9)
The autocovariance function of X(t) is defined by
KXV, s) = CovCX(t), X(s)l = E{CX(t) - px(t)lCX(s) - Px(s)l>
= Rx(t, s) - Px(t)Px(s) (5.1 0)
It is clear that if the mean of X(t) is zero, then Kx(t, s) = Rx(t, s). Note that the variance of X(t) is
given by

If X(t) is a complex random process, then its autocorrelation function Rx(t, s) and autocovariance
function Kx(t, s) are defined, respectively, by

5.4 CLASSIFICATION OF RANDOM PROCESSES
If a random process X(t) possesses some special probabilistic structure, we can specify less to
characterize X(t) completely. Some simple random processes are characterized completely by only the
first- and second-order distributions.