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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 17 Today: Random Processes: (1) Expectation (2) Correlation & Covariance (3) Wide Sense Stationarity 1 Expectation of Random Processes 1.1 Expected Value and Correlation Defn: Expected Value of a Random Process The expected value of continuous time random process X ( t ) is the deterministic function X ( t ) = E X ( t ) [ X ( t )] for the discretetime random process X n , X [ n ] = E X n [ X n ] Example: What is the expected value of a Poisson process? Let Poisson process X ( t ) have arrival rate . We know that X ( t ) = E X ( t ) [ X ( t )] = summationdisplay x =0 x ( t ) x x ! e t = e t summationdisplay x =0 x ( t ) x x ! = e t summationdisplay x =1 ( t ) x ( x 1)! = ( t ) e t summationdisplay x =1 ( t ) x 1 ( x 1)! = ( t ) e t summationdisplay y =0 ( t ) y ( y )! = ( t ) e t e t = t (1) This is how we intuitively started deriving the Poisson process  we said that it is the process in which on average we have arrivals per unit time. Thus wed certainly expect to see t arrivals after a time duration t . ECE 5510 Fall 2008 2 Example: What is the expected value of X n , the number of successes in a Bernoulli process after n trials? We know that X n is Binomial, with mean np . This is the mean function, if we consider it to be a function of n : X [ n ] = E X [ X n ] = np . 1.2 Autocovariance and Autocorrelation These next two definitions are the most critical concepts of the rest of the semester. Generally, for a timevarying signal, we often want to know two things: How to predict its future value. (It is not deterministic.) WeHow to predict its future value....
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This note was uploaded on 09/15/2011 for the course ECE 5510 taught by Professor Chen,r during the Fall '08 term at University of Utah.
 Fall '08
 Chen,R

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