not a valid random variable

• Oct 31st 2013, 11:49 AM
awais100
not a valid random variable
Give an example for a mapping from the sample space to R that is not a valid random variable.
• Oct 31st 2013, 12:29 PM
SlipEternal
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?
• Oct 31st 2013, 12:39 PM
awais100
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...
• Oct 31st 2013, 01:26 PM
SlipEternal
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.