ECE313.Lecture38

ECE313.Lecture38 - Expectation, Covariance, and Correlation...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications ECE 313 - Lecture 38 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, 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 Urbana-Champaign, 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 Urbana-Champaign, 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 Urbana-Champaign, 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 n-dimensional integral of u i f X (u ) over the entire space or the n-fold 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 Urbana-Champaign, 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 (n-dimensional) integral of g(u i )f X (u ) over the entire space or the n-fold 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 Urbana-Champaign, All Rights Reserved Slide 7 of 36 Joint pdf f X (u ) or pmf p X (u ) g( X ) = g( X 1 , X 2 ,,...
View Full Document

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.

Page1 / 38

ECE313.Lecture38 - Expectation, Covariance, and Correlation...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online