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6 thus to understand the likelihood of having a

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Unformatted text preview: X (6) = 3. More succinctly, we might write X : Ω → R deﬁned by X (ω ) = ω − 3. The function X is an example of a random variable. Observe that 1 P {X = −2} = P {ω ∈ Ω : X (ω ) = −2} = P {1} = , 6 and, similarly, 1 P {X = −1} = P {X = 0} = P {X = 1} = P {X = 2} = P {X = 3} = . 6 Thus, to understand the likelihood of having a certain net winning, it is enough to know the probabilities of the outcomes associated with that net winning. This leads to the general notion of a random variable as a real-valued function on Ω. As we will see shortly, the sort of trouble that we had with constructing the uniform probability on the uncountable space ([0, 1], B1 ) is the same sort of trouble that will prevent any realvalued function on Ω from being a random variable. It will turn out that only a special type of function, known as a measurable function, will be a random variable. Fortunately, every reasonable function (including those that one is likely to encounter when applying probability theory to everyday chance experiments such as casino games) will be measurable. For a function not to be measurable, it will need to be really weird. The deﬁnition of random variable Suppose that (Ω, F , P) is a probability space. As in the example above, we want to compute probabilities associated with certain values of the random variable; that is, we want to compute P {X ∈ B } for any Borel set B . Hence, if we want...
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This document was uploaded on 04/07/2014.

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