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I do not understand the relation of a normal variable "x", which is to me just a placeholder for an element of a set, and a random variable, which is a mapping from the set of all possible events to $\mathbb{R}$.

To make the question more concrete, some parts I am struggling with:

  • Normal functions with variables can be evaluated, e.g. $f(x)=2x$ will plot to a line, can I do this with a function of a random variable?
  • Furthermore the randomness part seems to be missing, why is it not neccessary to define how the random variable obtains its random value?

Edit: now that I feel I understood the definition, I cannot say what exacly what was missing for understanding. Nevertheless reading the following links, the comments and answer here, did the trick.

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    There was a random process of picking up names for mathematical objects, and that's what came up. What are the odds of that happening? :-) – Asaf Karagila Feb 23 '14 at 19:02
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    Can you give more details about what exactly confuses you in the linked questions? – M Turgeon Feb 23 '14 at 19:04
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    You can do this with a random variable too (but please dont expect to get the same figure all the time!) AND when you look at a probability space you will see that no information is missing. – Seyhmus Güngören Feb 23 '14 at 19:20
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    @AsafKaragila: You are implying that the process of name-picking is one of selection with replacement. Which sadly is true. – Marc van Leeuwen Feb 27 '14 at 08:46
  • Unrelated question, but why is the $\omega$ omitted? I.e, why is it so customary to write $X$ for the "value" of $X$ instead of $X(\omega)$? – rubikscube09 Feb 11 '19 at 22:01

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I'm going to give a very unsophisticated answer, as I don't see how to improve on the more sophisticated answers given in the links.

Let's say we pick a cat at random from the population of all cats in the world, and measure its weight, call it $W$. Then $W$ is a random variable. I hope it's clear why we'd use that term for it: it's a variable, and the cat was chosen at random.

The space $\Omega$, as you know, is the space of all outcomes to the "select a cat" experiment, so an element $c\in\Omega$ represents the outcome of selecting a particular cat. So $W$ depends on $c$: $W$ acquires a value once we've selected our cat. In other words, $W$ is a function of $c$. That is, $W:\Omega\rightarrow \mathbb{R}$.

The probability aspect appears when we ask about certain events. An event is a set of outcomes, for example, one event would be "the weight of the selected cat is between 2 and 3 kg". So the event is defined as $E=\{c\in\Omega | 2\text{kg}\leq W(c)\leq 3\text{kg}\}$. The probability that this occurs is $\mathcal{P}(E)$. The "random variable" is involved only in defining the set $E$; the probability measure takes it from there.

Michael Weiss
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    Let me suggest to rename $W$ your $w$. – Did Feb 24 '14 at 07:34
  • Let me suggest to rename $\omega$ your $e$ (previously named $\omega$, why did you change?). – Did Feb 27 '14 at 08:56
  • I changed it again, to $c$ (for cat). Why do you think it should be $\omega$? I looked through a few sources (e.g., Feller), and there does not seem to be any standard typographical convention for the points of the sample space. – Michael Weiss Feb 27 '14 at 18:27
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    I am ready to bet that most recent probability textbooks of mathematical leaning use small-omega to denote the running element of the set Capital-Omega, like one would use small-x for elements of Capital-X, and so on. This is pure notational logic, no? Unrelated: if W(c) is a real number, one can ask that W(c) is between 2 and 3, not between 2kg and 3kg. – Did Feb 27 '14 at 18:34