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Given a probability space $(\Omega, \mathcal{F}, P)$, we define a real-valued random variable $X$, which is a measurable function from $(\Omega, \mathcal{F})$ to $(R,\mathcal{B}(R))$.

Since $X$ is a function, what's the precise definition of one realization of $X$?

When I use Matlab to simulate one realization of $X$, I am actually using pseudo-random sequences, which are deterministic but statistically very close to the real distribution of $X$.

Is the realization of $X$ only an approximative definition? Can we ever really have one true realization of $X$?

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The difference between a random variable $X$ and a "realization" of it is the difference between a distribution and a sample from that distribution. In particular, a random variable $X$ is "formalized" in terms of a function from the sample space to some result space, typically $\mathbf{R}$. The realization of a random variable is "what you get" when an experiment is run, and you figure out which events happened, and you apply $X$ to those events.

That said, it's called a random variable because you can treat the sample from the experiment as a "hidden parameter" and identify $X$ with $X(\omega)$, by treating $\omega$ as arbitrary.

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  • $\begingroup$ Sorry my question may be too vague. In fact I am asking if there is a mathematical definition for "realization". Or a mathmetical explanation for where randomness comes from. Since when we use computer to sample, everything is in fact deterministic $\endgroup$ – Petite Etincelle May 28 '14 at 16:49
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    $\begingroup$ Whether your sampling on a computer is "random" or not isn't relevant to what the realization of a random variable is. A random variable is a bit of algebra, and the algebra will work no matter how you sample. If you want true non-determinism, you can use dice or a physical random number generator. $\endgroup$ – nomen May 28 '14 at 16:54

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