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Unformatted text preview: cases later on) that this converges to E X . When Ω is countable, there is an equivalent deﬁnition of expectation which is also useful. Proposition 3.1 E X = X ω ∈ Ω X ( ω ) P ( ω ) . Proof. E X = X x P ( X = x ) = X x x X { ω : X ( ω )= x } P ( ω ) = X x X { ω : X ( ω )= x } X ( ω ) P ( ω ) = X ω X ( ω ) P ( ω ) . Let f : R → R . 6...
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This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.
 Spring '10
 ansan

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