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The formal definition of a random variable is

Random variable over a sample space is a function from sample space to $R$

I want to get intuitively what actually it is doing. In this context, I had the following idea

A random variable is a method to search for actual input numbers (real numbers) for the probability measure function. Since sample space is a set of outcomes of a random experiment, they can be represented by any symbols and assigns a probability for such outcomes. In order to get the real numbers instead of symbols outcomes, we use random variables so that probability measure becomes a real function. So there are good random variables and bad random variables.

Is it correct? Are random variable for making probability measure function a real valued function?

From comments I came to know that there may be no superiority of a random variable over another, if it is the case, then is it true that for a particular sample space, there is no particular assignment of outcomes to real numbers that is desired inputs (for probability measure function) or useful in some way?

hanugm
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    You are missing measurable function. – b00n heT May 22 '19 at 08:52
  • I disagree with the conclusion "So there are good random variables and bad random variables." –  May 22 '19 at 08:53
  • "So there are good [...] and bad random vairables" seems unjustified – Hagen von Eitzen May 22 '19 at 08:53
  • @YvesDaoust I mean ranking among random variables, those provide good mapping (to actual real numbers) and others ... – hanugm May 22 '19 at 09:02
  • @HagenvonEitzen – hanugm May 22 '19 at 09:03
  • Maybe you refer to quantitative vs. qualitative values ? (say indexes assigned to colors) –  May 22 '19 at 09:04
  • @YvesDaoust didn't understand. But I mean that we can't treat all random variables on the same sample space equally. Suppose if random variable is a constant function, then it may not be a good random variable at all. – hanugm May 22 '19 at 09:10
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    I don't think this is a problematic issue. In general, the definition of the r.v. comes naturally. –  May 22 '19 at 09:17
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    @hanug, There's nothing wrong with a constant random variable. If there were, we couldn't say the random variables are closed under addition. – J.G. May 22 '19 at 09:58
  • Often in applications we want to estimate or predict the value of some quantity such as the selling price of a house. We don't know the value of this quantity, but we do know that some values are more likely than other values. So this quantity is a random variable. – littleO May 22 '19 at 10:13

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