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lecture14

# lecture14 - ECE 5510 Random Processes Lecture Notes Fall...

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ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 14 Today: (1) Decorrelation Transform (continued) (2) Joint Gaus- sian R.V.s HW 6 is due today; HW 7 is due a week from today, at 5pm. Discussion item? 0.1 Linear Transforms of R.V.s Continued Example: Average and Difference of Two i.i.d. R.V.s We represent the arrival of two people for a meeting as r.v.s X 1 and X 2 . Let X = [ X 1 , X 2 ] T . Assume that the two people arrive independently, with the same variance σ 2 and mean μ . Consider the average arrival time Y 1 = ( X 1 + X 2 ) / 2, and the difference between the arrival times, Y 2 = X 1 - X 2 . The latter is a wait time that one person must wait before the second person arrives. Show that the average time and the difference between the times are uncorrelated. You can take these steps to solve this problem: 1. Let Y = [ Y 1 , Y 2 ] T . What is the transform matrix A in the relation Y = A X ? 2. What is the mean matrix μ Y = E Y [ Y ]? (Note this isn’t really needed to answer the question, but is good practice anyway.) 3. What is the covariance matrix of Y ? 4. How does the covariance matrix show that the two are uncor- related? 1 Joint Gaussian r.v.s We often (OFTEN) see joint Gaussian r.v.s. E.g. ECE 5520, Digi- tal Communications, joint Gaussian r.v.s are everywhere. In addi- tion, the joint Gaussian R.V. is extremely important in statistics, economics, other areas of engineering. We can’t overemphasize its importance. In many areas of the sciences, the Gaussian r.v. is an approximation. In digital communications, control systems, and

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lecture14 - ECE 5510 Random Processes Lecture Notes Fall...

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