Suppose that x r is a simple random variable so that

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Unformatted text preview: ne E(X ) for positive random variables, (3) define E(X ) for general random variables. This strategy is sometimes called the “standard machine” and is the outline that we will follow to prove most results about expectation of random variables. Step 1: Simple Random Variables Let (Ω, F , P) be a probability space. Suppose that X : Ω → R is a simple random variable so that m ￿ aj 1 A j ( ω ) X (ω ) = j =1 where a1 , . . . , am ∈ R and A1 , . . . , Am ∈ F . We define the expectation of X to be E(X ) = m ￿ j =1 aj P {Aj } . Step 2: Positive Random Variables 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.) We are also assuming at this step that X is not a simple random variable. 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 }. (18.1) Proposition 18.1. For every random variable X ≥ 0, there exists a sequence (Xn ) of positive, simple random variables with Xn...
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