MIT2_854F10_covnotes

MIT2_854F10_covnotes - XY ] E ( X ) E ( Y ) E ( Y ) E ( X )...

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X, Y independent cov( X, Y ) = 0 Assume random variables X and Y are discrete . That is, assume that there is a fnite or denumerable sample space which is a set oF i and a set oF quantities x i and y i defned. Defnition X and Y are independent iF prob(( X = x ) and ( Y = y )) = prob( X = x )prob( Y = y ) in which x is some x i and y is some y j . Then iF X and Y are independent, E ( XY ) = E ( X ) E ( Y ) Proof: E ( XY ) = x i y j prob( XY = x i y j ) i,j = x i y j prob(( X = x i ) and ( Y = y j )) i,j = x i y j prob( X = x i )prob( Y = y j ) i,j = x i prob( X = x i ) y j prob( Y = y j ) = E ( X ) E ( Y ) i j Then iF X and Y are independent, cov( X, Y ) = E [( X E ( X ))( Y E ( Y ))] = E [ XY XE ( Y ) Y E ( X ) + E ( X ) E ( Y )] = E [
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Unformatted text preview: XY ] E ( X ) E ( Y ) E ( Y ) E ( X ) + E ( X ) E ( Y ) = 0 James Zhang Stan Gershwin MIT OpenCourseWare http://ocw.mit.edu 2.854 / 2.853 Introduction to Manufacturing Systems Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT2_854F10_covnotes - XY ] E ( X ) E ( Y ) E ( Y ) E ( X )...

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