Therefore using theorem 203 we conclude ey ex 12 213

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Unformatted text preview: r the expectations of discrete and continuous random variables encountered in introductory probability follow from the general theory developed. The first examples we will discuss, however, are the calculations of expectations directly from the definition and theory. Example 21.3. Consider ([0, 1], B1 , P) where B1 are the Borel sets of [0, 1] and P is the uniform probability. Let X : Ω → R be given by X (ω ) = ω so that X is a uniform random variable. We will now compute E(X ) directly by definition. Note that X is not simple since the range of X , namely [0, 1], is uncountable. However, X is positive. This means that as a consequence of Propositions 18.1 and 18.2, if Xn is a sequence of positive, simple random variables with Xn ↑ X , then E(Xn ) ↑ E(X ). Thus, let ￿ 0, 0 ≤ ω < 1/2, X1 (ω ) = 1/2, 1/2 ≤ ω ≤ 1 0 ≤ ω ≤ 1/4, 0, 1/4, 1/4 ≤ ω < 1/2, X2 (ω ) = 1/2, 1/2 ≤ ω < 3/4, 3/4, 3/4 ≤ ω ≤ 1, 21–2 and in general, Xn (ω ) = 2n − 2 ￿ j =1 j−1 1 2n ￿ j−1 j ≤ω< n n 2 2 ￿ 2n − 1 + 1 2n ￿ 2n − 2 2n − 1 ≤ω≤ 2n 2n ￿ . Thus...
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This document was uploaded on 04/07/2014.

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