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lecture11 - ECE 5510 Random Processes Lecture Notes Fall...

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ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 11 Today: Joint r.v.s (1) Expectation of Joint r.v.s, (2) Covari- ance (both in Y&G 4.7) HW 5 due Tue after break at 5pm in the HW locker. Office hours: Today after class to 1:15, and Thu 1-3. By the end of these lecture notes, we will have covered 4.1-4.5, 4.7-4.10; and 5.1-5.4, 5.6. For the lecture 12, read 4.6 and 4.11, and 5.5. 1 Expectation of Joint r.v.s We can find the expected value of a random variable or a function of a random variable similar to how we learned it earlier. However, now we have a more complex model, and the calculation may be more complicated. Def’n: Expected Value (Joint) The expected value of a function g ( X 1 , X 2 ) is given by, 1. Discrete: E X 1 ,X 2 [ g ( X 1 , X 2 )] = X 1 S X 1 X 2 S X 2 g ( X 1 , X 2 ) P X 1 ,X 2 ( x 1 , x 2 ) 2. Continuous: E X 1 ,X 2 [ g ( X 1 , X 2 )] = integraltext X 1 S X 1 integraltext X 2 S X 2 g ( X 1 , X 2 ) P X 1 ,X 2 ( x 1 , x 2 ) Essentially we have a function of two random variables X 1 and X 2 , called Y = g ( X 1 , X 2 ). Remember when we had a function of a random variable? We could either 1. Derive the pdf/pmf of Y, and then take the expected value of Y . 2. Take the expected value of g ( X 1 , X 2 ) using directly the model of X 1 , X 2 . Example: Expected Value of g ( X 1 )
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ECE 5510 Fall 2009 2 Let’s look at the discrete case: E X 1 ,X 2 [ g ( X 1 )] = summationdisplay X 1 S X 1 summationdisplay X 2 S X 2 g ( X 1 ) P X 1 ,X 2 ( x 1 , x 2 ) = summationdisplay X 1 S X 1 g ( X 1 ) summationdisplay X 2 S X 2 P X 1 ,X 2 ( x 1 , x 2 ) = summationdisplay X 1 S X 1 g ( X 1 ) P X 1 ( x 1 ) = E
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