L17_Sum_Exp_print

# L17_Sum_Exp_print - illustrates on our example what each of...

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Illustration of the Proof of Lemma 5.9 Version 2.1: Last updated, Nov 24, 2007

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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 } 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 } On the next page, the Right Hand Side lists the formulas that appeared in the proof of Lemma 5.9. The Left Hand Side

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Unformatted text preview: illustrates, on our example, what each of those formulas means. X s ∈ S X ( s ) P ( s ) = X ( HHH ) P ( HHH ) | {z } + X ( HHT ) P ( HHT ) + X ( HTH ) P ( HTH ) + X ( THH ) P ( THH ) | {z } + X ( HTT ) P ( HTT ) + X ( TTH ) P ( TTH ) + X ( THT ) P ( THT ) | {z } + X ( TTT ) P ( TTT ) | {z } F 1 F 2 F 3 F 4 = · P ( HHH ) | {z } + 1 ` P ( HHT ) + P ( HTH ) + P ( THH ) ´ | {z } + 2 · ` P ( HTT ) + P ( TTH ) + P ( THT ) ´ | {z } + 3 · P ( TTT ) | {z } F 1 F 2 F 3 F 4 = x 1 P ( F 1 ) + x 2 P ( F 2 ) + x 3 P ( F 3 ) + x 4 P ( F 4 ) = k X i =1 X s : s ∈ F i X ( s ) P ( s ) = k X i =1 x i X s : s ∈ F i P ( s ) = k X i =1 x i P ( F i ) = E ( X )...
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L17_Sum_Exp_print - illustrates on our example what each of...

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