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This is a concept question. The fundamental of modern probability theory is measure theory. A probability space is just a finite measure space and a random variable is just a measurable function. We know that for a fixed function $f$, $f(x)$ is unique for each $x$. How can a function be random? In what sense do we call a measurable function "random variable"?

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  • $\begingroup$ It's not random, but it models randomness. $\endgroup$ – Jair Taylor Aug 15 '15 at 3:01
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The output of a function is uniquely determined by its input. However, for a measurable function as a random variable, we don't know the input: only after a realization can we know $x$.

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The philosophical setup is that $\Omega$ is a sample space, $\mathscr{F}$ a collection of events that could happen. A measurable $X$ is called a random variable because we are pretending that we do not know which $\omega$ will occur. I am going to flip a coin and let $X$ be the indicator of heads. I model it as $X(\omega_1) = 1$ and $X(\omega_2) = 0$ (i.e. on $\omega_1$ I get heads and $\omega_2$ I get tails)

So will I get heads or tails? The point is if you don't know what $\omega \in \Omega$ will occur, then you can't answer the question. That's why we call it random.

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  • $\begingroup$ Thus every function is random. $\endgroup$ – Eduardo Longa Jul 27 '17 at 21:09
  • $\begingroup$ @EduardoLonga What if no $\sigma$-algebra or probability measure has been specified? $\endgroup$ – fourierwho Jul 28 '17 at 1:11

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