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I understand that the Random variable is a function such as,

$\text{RV}:\sigma-\text{field}(\Bbb C) \rightarrow \text{range of random variable} \in\Bbb R$ where $\Bbb C$ denotes sample space.

Is there any particular name for the range of random variable?

StubbornAtom
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Beverlie
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    The domain of a rv is usually not a $\sigma$-field. See here for a proper definition. I am not aware of any particular name for the range. Have you a specific reason to be curious about this? – drhab Jun 24 '17 at 08:38
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    @drhab good question. When I imagine the Sample Space, I eventually remember I need to construct the RV function before assign the element of Range of RV into Probablity value. To me, I am mostly emotionally interested in the course of assigning each element in sample space into Some kind of numerical values in range of RV. So I just want to name it for my convenience. – Beverlie Jun 24 '17 at 08:41
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    @drhab additionally, it looks very well close to my fundamental question, how to understand the mathematically precise mapping from possibile outcomes to numbers, but wikipedia looks little bit fast and difficult to me. Which book would be sufficiently enough to start and understand the measurable functions and outcomes and the construction of random variable and possibility? – Beverlie Jun 24 '17 at 08:45
  • Unfortunately I cannot reach unto you any useful resources. This mainly because I am very autodidactic myself and only pick out of books/scripts/internet what I really need (leaving the rest aside). Let's hope that someone else comes along who can help you better. Good luck with your study. – drhab Jun 24 '17 at 08:50

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As someone who took his first statistics class after his honors analysis class in college, I tried to think of a random variable as a real-valued, measurable function $X$ on a $\sigma$-field $\Omega$.

Although that's technically the definition of random variable, it doesn't capture how they are used. What helped me grok random variables is the notion that, for any interval $(a,b)$ in $\mathbb{R}$, we can find $$ P(a < X < b) = P\left(\left\{\omega \in \Omega\mid a < X(\omega) < b\right\}\right) $$

Matthew Leingang
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  • How did you know Ihad took analysis class..? Lol – Beverlie Jun 24 '17 at 08:55
  • What is measurable function? – Beverlie Jun 24 '17 at 08:57
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    I didn't say you have taken analysis, I said I had. And the advanced perspective (same as drhab's link to Wikipedia) on what a random variable “really was” didn't help me at all. That's why I brought it up; to say don't think of a RV as a function. Instead, think of it as something you can measure the probability that its values lie in any interval. – Matthew Leingang Jun 24 '17 at 09:01
  • Thx for your advice – Beverlie Jun 24 '17 at 09:02
  • What does this notation $P\left(\left{\omega \in \Omega\mid a < X(\omega) < b\right}\right)$ indicate? The probability that at least one $\omega$ exist? – robertspierre Jun 29 '19 at 13:46