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I'm curious what the most common way is to denote a random variable and the set of possible values that it can take on. I doesn't seem correct to say, for instance: $r.v.\ X \in \{\ldots\}$, because the random variable itself ($X$) is not a member of the set, the value of it takes on is.

I was also considering this: $$r.v.\ X=x;\quad x \in \{\ldots\}$$ Is that reasonable/understandable/common?

I've also seen random variables described as functions that map from a set of possible values into the reals: $$ X: \Omega \to \mathbb{R};\quad \Omega = \{\ldots\}$$ which is easy enough to denote, but I don't think it sufficiently conveys that $X$ is a random variables.

It seems like there should be some specific notation for "...takes on values from..." but I can't find any.

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I don't find the notation $X\in \{\ldots\}$ to be that terrible. The point is that $X$ is indeed a function, but one usually omits its argument. So you are in fact writing a condensed version of $$X(\omega)\in \{\ldots\},\quad \forall\omega \in \Omega. $$ –  Giuseppe Negro May 24 '13 at 13:20
    
Usually I just stay with the mapping definition. –  newbie May 24 '13 at 14:26
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