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131A_1_Lecture12-1_Winter_2012

# 131A_1_Lecture12-1_Winter_2012 - EE 131A Probability...

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UCLA EE131A (KY) 1 EE 131A Probability Professor Kung Yao Electrical Engineering Department University of California, Los Angeles Lecture 12-1

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UCLA EE131A (KY) 2 Conditional expectation (1) Review: For a rv X with a pdf f X (x), the expectation of X,E{X},or the expectation of g(X),E{g(X)}are known. However, in Lec11-2, we also know that a conditional pdf is also a pdf. Thus, we can consider the conditional expectation of a rv Y given X. 1. Consider X and Y to be discrete rv’s. The conditional expectation of Y given X = x k is defined by  jj k jk j = 1 kj j j=1 kk y P(Y=y ,X=x ) P(Y=y ,X=x ) EY|X=x = y P(Y=y|X=x ) y = . P(X=x )  
UCLA EE131A (KY) 3 Conditional expectation (2) 2. Let X and Y to be continuous rv’s. The conditional expectation of Y given X = x is defined by Ex. 1. (Ex. 2, p. 9, Lec11-2).  XY XY - Y|X XX -- y f( , y) dy , y ) EY |X=x = y y|x) dy = y d y = . f (x) f (x) x x   -x -y -y XY Y|X -x -x -x X x , y ) 2 e e e f (y|x)= ,0 y x< . f (x) 2e (1 e ) 1 e   -x -y -(x-y) XY X|Y -2y Y f (x,y) 2 e e x |y)= e y x < y ) 2 e 2, 0 , (, ) 0, elsewher .

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131A_1_Lecture12-1_Winter_2012 - EE 131A Probability...

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