Note that when we say let x be a random variable we

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Unformatted text preview: to be able to compute P {X ∈ B } = P {ω : X (ω ) ∈ B } = P {X −1 (B )} for every Borel set B , then it must be the case that X −1 (B ) is an event (which is to say that X −1 (B ) ∈ F for every B ∈ B ). 11–2 Definition. A real-valued function X : Ω → R is said to be a random variable if X −1 (B ) ∈ F for every Borel set B ∈ B . Note that when we say let X be a random variable, we really mean let X be a function from the probability space (Ω, F , P) to the real numbers endowed with the Borel σ -algebra (R, B ) such that X −1 (B ) ∈ F for every B ∈ B . Hence, when we define a random variable, we should really also state the underlying probability space as the domain space of X . Since every random variable we will consider is real-valued, our codomain (or target) space will always be R endowed with the Borel σ -algebra B . If we want to stress the domain space and codomain space, we will be explicit and write X : (Ω, F , P) → (R, B ). Example 11.2. Perhaps the simplest example of a random variable is the indicator function of an event. Let (Ω, F , P) be a probability space and suppose that A ∈ F is an event. Let X : Ω → R be given by ￿ 1, if ω ∈ A, X (ω ) = 1A (ω ) = 0, if ω ∈ A. / For B ∈ B , we find ∅, A, −1 (1A ) (B ) = Ac , Ω, if if if if 0 ￿∈ B , 0 ￿∈ B , 0 ∈ B, 0 ∈ B, 1 ￿∈ B, 1 ∈ B, 1 ￿∈ B, 1 ∈ B. Thus, since ∅, A, Ac , and Ω belong to F , we see that for any B ∈ B we necessarily have X −1 (B ) = (1A )−1 (B ) ∈ F proving that X is a random variable. 11–3...
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