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Unformatted text preview: Expectation, Covariance, and Correlation Professor Dilip V. Sarwate Department of Electrical and Computer Engineering 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign. All Rights Reserved ECE 313 Probability with Engineering Applications ECE 313  Lecture 38 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 2 of 36 The expected values of X and Y are Expectation from joint pdfs/pmfs for continuous random variables and by E[ X ] = u i p X (u i ); E[ Y ] = v i p Y (v i ) for discrete random variables Given the joint pdf or pmf of X and Y , we can first compute the (marginal) pdf or pmf of X or Y and substitute in the above E[ X ] = uf X (u) du; E[ Y ] = vf Y (v) dv ECE 313  Lecture 38 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 3 of 36 Doing everything in one step f X (u) = f X , Y (u,v) dv; f Y (v) = f X , Y (u,v) du v= u= E[ X ] = uf X (u) du; E[ Y ] = vf Y (v) dv E[ X ] = u f X , Y (u,v) dv du = uf X , Y (u,v) dv du = uf X , Y (u,v) du dv ECE 313  Lecture 38 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 4 of 36 It works the same way for Y too! E[ Y ] = v f X , Y (u,v) du dv = vf X , Y (u,v) du dv E[ X ] = uf X , Y (u,v) du dv Similarly, = vf X , Y (u,v) dv du For discrete RVs, integrals become sums ECE 313  Lecture 38 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 5 of 36 Given the joint pdf f X (u ) or pmf p X (u ) of n random variables X 1 , X 2 , X n , E[ X i ] is given by the ndimensional integral of u i f X (u ) over the entire space or the nfold sum of u i p X (u ) Expectation of a vector If X = ( X 1 , X 2 , X n ), then E[ X ] = (E[ X 1 ], E[ X 2 ], E[ X n ]) Generalization to n variables ECE 313  Lecture 38 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 6 of 36 Joint pdf f X (u ) or pmf p X (u ) of n random variables X 1 , X 2 , X n E[g( X i )] is given by the (ndimensional) integral of g(u i )f X (u ) over the entire space or the nfold sum of g(u i )p X (u ) This is just LOTUS with the calculation of the marginal pdf of X i being merged with the calculation of E[g( X i )] into one giant step for mankind LOTUS works in the same way ECE 313  Lecture 38 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, All Rights Reserved Slide 7 of 36 Joint pdf f X (u ) or pmf p X (u ) g( X ) = g( X 1 , X 2 ,,...
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This note was uploaded on 09/29/2009 for the course ECE 123 taught by Professor Mr.pil during the Spring '09 term at University of Iowa.
 Spring '09
 mr.pil

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