Page 179 - Schaum's Outlines - Probability, Random Variables And Random Processes

RANDOM PROCESSES [CHAP 5

5.7 WIENER PROCESSES
Another example of random processes with independent stationary increments is a Wiener process.

DEFINITION 5.7.1
A random process (X(t), t 2 0) is called a Wiener process if

1. X(t) has stationary independent increments.
2. The increment X(t) - X(s) (t > s) is normally distributed.
3. E[X(t)J =O.
4. X(0) = 0.

The Wiener process is also known as the Brownian motion process, since it originates as a model for
Brownian motion, the motion of particles suspended in a fluid. From Def. 5.7.1, we can verify that a
Wiener process is a normal process (Prob. 5.61) and

where a2 is a parameter of the Wiener process which must be determined from observations. When
a2 = 1, X(t) is called a standard Wiener (or standard Brownian motion) process.
The autocorrelation function Rx(t, s) and the autocovariance function K,(t, s) of a Wiener
process X(t) are given by (see Prob. 5.23)

DEFINITION 5.7.2
A random process (X(t), t 2 0) is called a Wiener process with drift coeficient p if
1. X(t) has stationary independent increments.
2. X(t) is normally distributed with mean pt.
3. X(0) = 0.

From condition 2, the pdf of a standard Wiener process with drift coefficient p is given by
4

Solved Problems

RANDOM PROCESSES
5.1. Let XI, X,, . . . be independent Bernoulli r.v.'s (Sec. 2.7A) with P(X, = 1) = p and P(X, = 0) =
q = 1 - p for all n. The collection of r.v.'s (X,, n 2 1) is a random process, and it is called a
Bernoulli process.
(a) Describe the Bernoulli process.
(b) Construct a typical sample sequence of the Bernoulli process.
(a) The Bernoulli process {X,, n 2 1) is a discrete-parameter, discrete-state process. The state space is
E = (0, I), and the index set is T = {1,2, . . .).