{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture17

# lecture17 - ECE 5510 Random Processes Lecture Notes Fall...

This preview shows pages 1–3. Sign up to view the full content.

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 Def’n: Expected Value of a Random Process The expected value of continuous time random process X ( t ) is the deterministic function μ X ( t ) = E X ( t ) [ X ( t )] for the discrete-time random process X n , μ 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 λ . We know that μ X ( t ) = E X ( t ) [ X ( t )] = summationdisplay x =0 x ( λt ) x x ! e - λt = e - λt summationdisplay x =0 x ( λt ) x x ! = e - λt summationdisplay x =1 ( λt ) x ( x - 1)! = ( λt ) e - λt summationdisplay x =1 ( λt ) x - 1 ( x - 1)! = ( λt ) e - λt summationdisplay y =0 ( λt ) y ( y )! = ( λt ) e - λt e λt = λt (1) This is how we intuitively started deriving the Poisson process - we said that it is the process in which on average we have λ arrivals per unit time. Thus we’d certainly expect to see λt arrivals after a time duration t .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 : μ 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 time-varying signal, we often want to know two things: How to predict its future value. (It is not deterministic.) We will use the ‘autocovariance’ to determine this.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

lecture17 - ECE 5510 Random Processes Lecture Notes Fall...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online