Stochastic Processes Dr. Talal Skaik Chapter 10 1 Probability and Stochastic Processes A friendly introduction for electrical and computer engineers Electrical Engineering department Islamic University of Gaza December 2011

The word stochastic means random. The word process in this context means function of time. 2

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Example: where is a uniformly distributed random variable in represents a stochastic process. 4

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Ensemble average: With t fixed at t=t 0, X(t 0 ) 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, s 0 ), and produces a typical number for this sample function. 6

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14 For a specific t, X(t) is a random variable with distribution:

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When Cov[X,Y] is applied to two random variables that are observations of X(t) taken at two different times, t 1 and t 2 =t 1 + τ seconds:  The covariance indicates how much the process is likely to change in the τ seconds elapsed between t 1 and t 2.  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. Autocovariance 20

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 Recall in a stochastic process X(t), there is a random variable X(t 1 ) at every time t 1 with PDF f X(t1) (x).  For most random processes, the PDF f X(t1) (x) depends on t 1.  For a special class of random processes know as stationary processes, f X(t1) (x) does not depend on t 1.  Therefore: the statistical properties of the stationary process do not change with time (time-invariant). 23

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