L18_Prod_Exp_print

L18_Prod_Exp_print - ( Z = 1) + 2 · P ( Z = 2) + 4 · P (...

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Illustration of the Proof of Lemma 5.28 Version 2.1: Last updated, Nov 24, 2007
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In class, we proved that the expectation of the product of two independent random variables, is the product of their expectations . Formally In these sides, we illustrate the proof of Lemma 5.28 with an example . Lemma 5.28 If X and Y are independent random variables on sample space S with values x 1 ,x 2 ,...,x k and y 1 ,y 2 ,...,y m , respectively, then E ( XY ) = E ( X ) E ( Y ) .
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P ( Z = 1) = P ( X = 1 Y = 1) = 1 6 Suppose that we have two independent random variables, X,Y that each can take on the values 1 , 2 , 4 , but with different probability weights. P ( X = 1) = 1 / 3 P ( X = 2) = 1 / 3 P ( X = 4) = 1 / 3 P ( Y = 1) = 1 / 2 P ( Y = 2) = 1 / 4 P ( Y = 4) = 1 / 4 E ( X ) = 7 / 3 E ( Y ) = 2 E ( X )( EY ) = 14 / 3 Z = XY can only take on the values 1 , 2 , 4 , 8 , 16 . P ( Z = 2) = P ( X = 1 Y = 2)+ P ( X = 2 Y = 1) = 1 12 + 1 6 = 1 4 P ( Z = 4) = P ( X = 1 Y = 4)+ P ( X = 4 Y = 1) + P ( X = 2 Y = 2) = 1 12 + 1 6 + 1 12 = 1 3 P ( Z = 8) = P ( X = 2 Y = 4)+ P ( X = 4 Y = 2) = 1 12 + 1 12 = 1 6 P ( Z = 16) = P ( X = 4 Y = 4) = 1 12
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Z = XY can only take on the values 1 , 2 , 4 , 8 , 16 . So = 1 · P
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Unformatted text preview: ( Z = 1) + 2 · P ( Z = 2) + 4 · P ( Z = 4) + 8 · P ( Z = 8) + 16 · P ( z = 16) = 14 3 = E ( X ) · E ( Y ) E ( XY ) = E ( Z ) On the next page, we mimic the proof of Lemma 5.28, using these X,Y . Reading the proof with this example in mind, might make the proof more understandable. E ( X ) E ( Y ) = X x ∈{ 1 , 2 , 4 } xP ( X = x ) X y ∈{ 1 , 2 , 4 } yP ( Y = y ) = X x ∈{ 1 , 2 , 4 } X y ∈{ 1 , 2 , 4 } xyP ( X = x ) P ( Y = y ) = X z ∈{ 1 , 2 , 4 , 8 , 16 } z X x,y ∈{ 1 , 2 , 4 } xy = z P ( X = x ) P ( Y = y ) = X z ∈{ 1 , 2 , 4 , 8 , 16 } z X x,y ∈{ 1 , 2 , 4 } xy = z P ` ( X = x ) ∧ ( Y = y ) ´ = X z ∈{ 1 , 2 , 4 , 8 , 16 } zP ( Z = z ) = E ( Z ) = E ( XY ) Ind of X,Y...
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This note was uploaded on 08/25/2010 for the course COMP COMP170 taught by Professor M.j.golin during the Spring '10 term at HKUST.

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L18_Prod_Exp_print - ( Z = 1) + 2 · P ( Z = 2) + 4 · P (...

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