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Probability and Stochastic Processes
A friendly introduction for electrical and computer engineers
Chapter 10
Stochastic Processes
Dr. Talal Skaik
Electrical Engineering department
Islamic University of Gaza
December 2011
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•The word stochastic means random.
•The word process in this context means function of time.
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Example: X (t )  a cos( 0 t   ), where
random variable in
(0, 2 ),

is a uniformly distributed
represents a stochastic process.
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Ensemble average:
With t fixed at t=t0, X(t0) is a random variable, we have the
averages ( expected value and variance) as we studied earlier.
Time average:
applies to a specific sample function x(t, s0), and produces a
typical number for this sample function.
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For a specific t, X(t) is a random variable with distribution:
F ( x, t )
f ( x, t ) 
F ( x, t )  p[ X (t )  x]
x
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Autocovariance
When Cov[X,Y] is applied to two random variables that are
observations of X(t) taken at two different times, t1 and t2
=t1 +τ seconds:
The covariance indicates how much the process is likely
to change in the τ seconds elapsed between t1 and t2.
A high covariance indicates that the sample function is
unlikely to change much in the τ-second interval.
A covariance near zero suggests rapid change.
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Recall in a stochastic process X(t), there is a random
variable X(t1) at every time t1 with PDF fX(t1)(x).
For most random processes, the PDF fX(t1)(x) depends on t1.
For a special class of random processes know as
stationary processes, fX(t1)(x) does not depend on t1.
Therefore: the statistical properties of the stationary process
do not change with time (time-invariant).
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