Lect12_2700_s09

Lect12_2700_s09 - Review Independence ≥ 2 Conditional...

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Unformatted text preview: Review: Independence ≥ 2 Conditional Expectation Covariance Title Page JJ II J I Page 1 of 32 Go Back Full Screen Close Quit ENGRD 2700 Basic Engineering Probability and Statistics Lecture 12: Multivariate Distributions David S. Matteson School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA [email protected] March 3, 2009 Review: Independence ≥ 2 Conditional Expectation Covariance Title Page JJ II J I Page 2 of 32 Go Back Full Screen Close Quit 1. Review: Review: Independence ≥ 2 Conditional Expectation Covariance Title Page JJ II J I Page 3 of 32 Go Back Full Screen Close Quit Recall p ( x, y ) satisfies 1. p ( x, y ) ≥ , 2. ∑ { ( x,y ) ∈ possible values } p ( x, y ) = 1 . Get the marginal mass function by summing over the other variable: p X ( x ) = ∑ y p ( x, y ); p Y ( y ) = ∑ x P ( x, y ) Reason: [ X = x ] = { s ∈ S : X ( s ) = x } ∩ S = { s ∈ S : X ( s ) = x } ∩ [ y { s : Y ( s ) = y } = [ y { s ∈ S : X ( s ) = x,Y ( s ) = y } = [ y [ X = x,Y = y ] . Take probabilities and use the fact that the union is a disjoint union so the probabiity of a union is the sum of the probabilities. Review: Independence ≥ 2 Conditional Expectation Covariance Title Page JJ II J I Page 4 of 32 Go Back Full Screen Close Quit Review: continuous case Review: Independence ≥ 2 Conditional Expectation Covariance Title Page JJ II J I Page 5 of 32 Go Back Full Screen Close Quit Properties of joint density f ( x, y ): 1. f ( x, y ) ≥ , for all ( x, y ) ∈ R 2 ; 2. R ∞-∞ R ∞-∞ f ( x, y ) dxdy = 1 . Getting marginal densities: Integrate out the other variable: f X ( x ) = Z ∞ y =-∞ f ( x, y ) dy, f Y ( y ) = Z ∞ x =-∞ f ( x, y ) dx. Reason: F X ( x ) = P [ X ≤ x ] = P [ X ≤ x, Y ∈ R ] = Z x-∞ Z ∞-∞ f ( u, v ) dudv. Review: Independence ≥ 2 Conditional Expectation Covariance Title Page JJ II J I Page 6 of 32 Go Back Full Screen Close Quit Differentiate with respect to x on both sides to get f X ( x ) = Z ∞ v =-∞ f ( x, v ) dv. Review: Independence ≥ 2 Conditional Expectation Covariance Title Page JJ II J I Page 7 of 32 Go Back Full Screen Close Quit Example. Suppose F ( x, y ) = ( k ( x 2 + y 2 ) , if 0 ≤ x ≤ 2 , 1 ≤ y ≤ 4 , , otherwise. Good habit: Instead of specifying the function by cases, use indicator functions 1 A ( x ) = ( 1 , if x ∈ A, , if x / ∈ A....
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This note was uploaded on 03/05/2009 for the course ENGRD 2700 taught by Professor Staff during the Spring '05 term at Cornell.

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Lect12_2700_s09 - Review Independence ≥ 2 Conditional...

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