851_lectures17_24

# Note that x 0 and x 0 so that the positive part and

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Unformatted text preview: mple random variables with Xn ↑ X , then E(Xn ) ↑ E(X ). We will prove these facts next lecture. Now suppose that X is any random variable. Write X + = max{X, 0} and X − = − min{X, 0} for the positive part and the negative part of X , respectively. Note that X + ≥ 0 and X − ≥ 0 so that the positive part and negative part of X are both positive random variables and X = X + − X − and |X | = X + + X − . Deﬁnition. A random variable X is called integrable (or has ﬁnite expectation ) if both E(X + ) and E(X − ) are ﬁnite. In this case we deﬁne E(X ) to be E(X ) = E(X + ) − E(X − ). 17–3 Statistics 851 (Fall 2013) Prof. Michael Kozdron October 18, 2013 Lecture #18: Construction of Expectation Recall that our goal is to deﬁne E(X ) for all random variables X : Ω → R. We outlined the construction last lecture. Here is the summary of our strategy. Summary of Strategy for Constructing E(X ) for General Random Variables We will (1) deﬁne E(X ) for simple random variables, (2) deﬁ...
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