L17_Sum_Exp

L17_Sum_Exp - Illustration of the Proof of Lemma 5.9...

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Unformatted text preview: Illustration of the Proof of Lemma 5.9 Version 2.1: Last updated, Nov 24, 2007 The Expectation of Random Variable X is defined as E ( X ) = k X i =1 x i P ( X = x i ) . In class, we proved the equivalence of the following alternative method of calculating E ( X ) Lemma 5.9 If a random variable X is defined on a (finite) sample space S , then its expected value is given by E ( X ) = X s : s ∈ S X ( s ) P ( s ) . In these sides, we illustrate the proof of Lemma 5.9 with an example . Our example will be the space of 3 coin flips. X will be the number of heads in the flip S= { TTT, TTH, THT, HTT, THH, HTH, HHT, HHH } Our example will be the space of 3 coin flips. X will be the number of heads in the flip S= { TTT, TTH, THT, HTT, THH, HTH, HHT, HHH } Possible values for X are x 1 , = 0 , x 2 = 1 , x 3 = 2 , x 4 = 3 . We group the outcomes by their X values F 1 = ( X = x 1 ) = { HHH } F 2 = ( X = x 2 ) = { THH,HTH,HHT } F 3 = ( X = x 3 ) = { HTT,THT,TTH } F 4 = ( X = x 4 ) = { TTT } Our example will be the space of 3 coin flips. X will be the number of heads in the flip S= { TTT, TTH, THT, HTT, THH, HTH, HHT, HHH } Possible values for X are x 1 , = 0 , x 2 = 1 , x 3 = 2 , x 4 = 3 ....
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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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L17_Sum_Exp - Illustration of the Proof of Lemma 5.9...

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