131A_1_Monday_May_19_lecture

131A_1_Monday_May_19_lecture - Monday May 19th Lecture...

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UCLA EE131A (KY) 1 Monday May 19 th Lecture Conditional expectation Expectation of two rv’s Covariance of two rv’s Correlation coefficient of two rv’s Distribution and pdf of (X+Y) Central Limit Theorem
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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, 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 ) ∞∞ = ∑∑
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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. Consider the previous ex. {} 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 = -x -y XY 2 e c , 0 y x < , f (x,y) = 0, elsewhere . ≤∞
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UCLA EE131A (KY) 4 Conditional expectation (3) Ex. 1. (Continued). Find the conditional expectation of Y given X = x. Find the conditional expectation of X given Y = y. {} xx x x -y -x ud v 00 0 0 e1 1 1 E Y|X=x = y dy = y e dy uv - v du ye + e dy 1-e x 11 1 e x e = -xe -e -xe -e -1 , 0 x < 0 ⎫⎧ == ⎬⎨ ⎩⎭ ⎧⎫ −− ⎨⎬ ∫∫ . -(x-y) y y y u dv yy y y y- y - x y - y - y E X|Y=y = xe dx = e xe dx = e uv - v du = e + e dx = e ye -e = e ye e = 1 + y , 0 x . y ∞∞ ⎪⎪ +≤
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UCLA EE131A (KY) 5 Review: For a single rv X, we can define various expectations, such as: mean: μ = E{X}; n-th moment: m n = E{X n }; variance: σ 2 = E{(X- μ) 2 }; expectation of g(X) = Y: μ Y = E{Y} = E{g(X)}, etc. Now, we want to consider various joint expectations of X and Y using their joint pdf f XY (x,y) in the expectations. Expectation of two rv’s (1)
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UCLA EE131A (KY) 6 Expectation of two rv’s (2) Joint first moment μ XY of X and Y . Introduced concept of two conditional expectations {} XY X|Y Y f (x,y) Y Y y= x= E {X|Y=y} μ = E{XY} = xy f (x,y) dxdy = xy f (x|y)f (y) dxdy = y x f (x|y) dx f (y)dy = E Y E {X|Y=y} ∞∞ ←→ −∞ −∞ −∞ ⎧⎫ ⎨⎬ ⎩⎭ ∫∫ Y|X X X X f E {Y|X=x} (y|x)f (x) dxdy = x y f (y|x) dy f (x)dx = E X E {Y|X=x} . ⎪⎪ X|Y X|Y x= Conditional expectation of X given Y=y: E {X|Y=y} = x f (x|y) dx , Y|X Y|X Conditional expectation of Y given X=x: E {Y|X=x} = y f (y|x) dy .
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UCLA EE131A (KY) 7 Expectation of two rv’s (3) If X and Y are independent, then 1. The joint first moment 2. The conditional expectations become XY XY XY X Y -- μ = E{XY} = xy f (x,y) dxdy = x f (x) dx = y f (y) dy = μ μ ∞∞ −∞ −∞ ∫∫ X|Y X|Y X X x= x= E {X|Y=y} = x f (x|y) dx = x f (x) dx = E {X} , −∞ Y|X Y|X Y Y y= E {Y|X=x} = y f (y|x) dy = y f (y) dy = E {Y} .
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UCLA EE131A (KY) 8 Expectation of two rv’s (4) Ex. 1.
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This note was uploaded on 02/09/2011 for the course EE 131A taught by Professor Lorenzelli during the Spring '08 term at UCLA.

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131A_1_Monday_May_19_lecture - Monday May 19th Lecture...

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