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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 13 Today: (1) Covariance Matrices, (2) Gaussian R.V.s • AA 3 and HW 5 due at 5pm today. • HW 6 assigned today, due Tue, Oct. 27. • I will be out of town Tue, Nov.3 (and Wed, Nov 4) for a conference. We will not have class Tue. Nov 3. Consider it time to work on AA 4 (due Nov 3) and HW 7 (due Thu, Nov 5 at 10:45 am) • Discussion Item? 1 Covariance of a R.V. Def’n: Covariance Matrix The covariance matrix of an n-length random vector X is an n × n matrix C X with ( i,j )th element equal to Cov ( X i ,X j ). In vector notation, C X = E X bracketleftbig ( X- μ X )( X- μ X ) T bracketrightbig Example: For X = [ X 1 X 2 X 3 ] T C X = Var X 1 [ X 1 ] Cov ( X 1 ,X 2 ) Cov ( X 1 ,X 3 ) Cov ( X 2 ,X 1 ) Var X 2 [ X 2 ] Cov ( X 2 ,X 3 ) Cov ( X 3 ,X 1 ) Cov ( X 3 ,X 2 ) Var X 3 [ X 3 ] You can see that for two r.v.s, we’ll have just the first two rows and two columns of C X – this is what we put on the board when we first talked about covariance as a matrix. Note for n = 1, C X = σ 2 X 1 . 2 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.importance....
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- Fall '08