851_lectures17_24

# If x is a positive random variable then we dene ex

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Unformatted text preview: ition. If one of E(X + ) or E(X − ) is inﬁnite, then we can still deﬁne E(X ) by setting ￿ +∞, if E(X + ) = +∞ and E(X − ) &lt; ∞, E(X ) = −∞, if E(X + ) &lt; ∞ and E(X − ) = +∞. However, X is not integrable in this case. Deﬁnition. If both E(X + ) = +∞ and E(X − ) = +∞, then E(X ) does not exist. Remark. We see that the standard machine is really not that hard to implement. In fact, it is usually enough to prove a result for simple random variables and then extend that result to positive random variables using Propositions 18.1 and 18.2. The result for general random variables usually then follows by deﬁnition. 18–4 Statistics 851 (Fall 2013) Prof. Michael Kozdron October 21, 2013 Lecture #19: Expectation and Integration Deﬁnition. Let (Ω, F , P) be a probability space, and let X : Ω → R be a random variable. If X is a simple random variable, say X (ω ) = n ￿ ai 1 A i ( ω ) i=1 for a1 , . . . , an ∈ R and A1 , . . . , An ∈ F , then we deﬁne E(X ) = n ￿ i=1 ai P {Ai } . If X is a po...
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## This document was uploaded on 04/07/2014.

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