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Math Help - not a valid random variable

  1. #1
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    Dhahran
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    not a valid random variable

    Give an example for a mapping from the sample space to R that is not a valid random variable.
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  2. #2
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    Re: not a valid random variable

    What is "the sample space"? Is it just an arbitrary sample space X? What do you mean an example for a mapping that is not a valid random variable? Valid in what sense?

    From what I gather from your question, I assume you mean that X is some set, and \sigma(X) is the smallest \sigma-algebra containing X. You are looking for a map f:X \to \mathbb{R} such that there exists a set A \subseteq \mathbb{R} such that A is measurable, f^{-1}(A) is not in \sigma(X). Is that what you mean?
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  3. #3
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    Re: not a valid random variable

    yes, what you assumed is absolutely correct. I want an example for this. I think if I assume a set that is not a field or that does not contain all subsets of Omega (Whole set), may be then I can prove that mapping of this set to R will not result in a random variable...
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  4. #4
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    Re: not a valid random variable

    Define a relation R \subseteq \mathbb{R}\times \mathbb{R} by aRb if and only if b-a \in \mathbb{Q}. Show that R is an equivalence relation. Then, partition \mathbb{R} by R. Choose a representative from each partition where x_R is the representative chosen from the part containing x. Define a function f:\mathbb{R} \to \mathbb{R} by f(x) = x_R. That should do it.
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