851_lectures17_24

To that end suppose that x is a positive random

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Unformatted text preview: Y , then E(X ) ≤ E(Y ). Exercise 17.3. Prove the previous fact. 17–2 Having already deﬁned E(X ) for simple random variables, our goal now is to construct E(X ) in general. To that end, suppose that X is a positive random variable. That is, X (ω ) ≥ 0 for all ω ∈ Ω. (We will need to allow X (ω ) ∈ [0, +∞] for some consistency.) Deﬁnition. If X is a positive random variable, deﬁne the expectation of X to be E(X ) = sup{E(Y ) : Y is simple and 0 ≤ Y ≤ X }. That is, we approximate positive random variables by simple random variables. Of course, this leads to the question of whether or not this is possible. Fact. For every random variable X ≥ 0, there exists a sequence (Xn ) of positive, simple random variables with Xn ↑ X (that is, Xn increases to X ). An example of such a sequence is given by ￿ k if 2k ≤ X (ω ) &lt; n, n Xn ( ω ) = 2 n, if X (ω ) ≥ n. k+1 2n and 0 ≤ k ≤ n2n − 1, (Draw a picture.) Fact. If X ≥ 0 and (Xn ) is a sequence of si...
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