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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
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P {X = −2} = P {ω ∈ Ω : X (ω ) = −2} = P {1} = ,
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and, similarly,
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P {X = −1} = P {X = 0} = P {X = 1} = P {X = 2} = P {X = 3} = .
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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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