elemprob-page10 - equivalent. Proof. Starting with the rst...

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We have E X = 1( 1 4 ) + 2( 1 8 ) + 3( 1 16 + ··· = 1 4 h 1 + 2( 1 2 ) + 3( 1 4 ) + ··· i = 1 4 1 (1 - 1 2 ) 2 = 1 . Example. Suppose we roll a fair die. If 1 or 2 is showing, let X = 3; if a 3 or 4 is showing, let X = 4, and if a 5 or 6 is showing, let X = 10. What is E X ? Answer. We have P ( X = 3) = P ( X = 4) = P ( X = 10) = 1 3 , so E X = X x P ( X = x ) = (3)( 1 3 ) + (4)( 1 3 ) + (10)( 1 3 ) = 17 3 . We want to give a second definition of E X . We set E X = X ω S X ( ω ) P ( ω ) . Remember we are only working with discrete random variables here. In the example we just gave, we have S = { 1 , 2 , 3 , 4 , 5 , 6 } and X (1) = 3 ,X (2) = 3 ,X (3) = 4 ,X (4) = 4 ,X (5) = 10 ,X (6) = 10 , and each ω has probability 1 6 . So using the second definition, E X = 3( 1 6 ) + 3( 1 6 ) + 4( 1 6 ) + 4( 1 6 ) + 10( 1 6 ) + 10( 1 6 ) = 34 6 = 17 3 . We see that the difference between the two definitions is that we write, for example, 3 P ( X = 3) as one of the summands in the first definition, while in the second we write this as 3 P ( X = 1) + 3 P ( X = 2). Let us give a proof that the two definitions are equivalent. Proposition 4.1. If X is a discrete random variable and S is countable, then the two definitions are
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Unformatted text preview: equivalent. Proof. Starting with the rst denition, we have X x x P ( X = x ) = X x x X { : X ( )= x } P ( ) . This is because the set ( X = x ) is the union of the disjoint sets { } , where the union is over those for which X ( ) = x . Bringing the x inside the sum, and using the fact that X ( ) = x , our expression is equal to X x X { : X ( )= x } x P ( ) = X x X { : X ( )= x } X ( ) P ( ) . Since the union of { : X ( ) = x } over all possible values of x is just all , we then have X X ( ) P ( ) , which is the second denition. One advantage of the second denition is that linearity is easy. We write E ( X + Y ) = X S ( X ( ) + Y ( )) P ( ) = X [ X ( ) P ( ) + Y ( ) P ( )] = X X ( ) P ( ) + X Y ( ) P ( ) = E X + E Y. 10...
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