Why is the function $\Omega\rightarrow\mathbb{R}$ called a random variable?

Question

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?

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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.

• The concept of random variable
• Definition of a real-valued random variable
• Understanding the definition of a random variable
• Random Variable
• Definition of random variable, Borel $\sigma$-algebra
• What exactly is a random variable?

Answer 1

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.