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Is there standard notation for sampling a value from a probability distribution? Like, if I had a random variable $X$, setting $x$ to whatever value I happened to sample from $X$ on this occasion? I was thinking just $x \gets X$, but that seems ambiguous.

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Use $X(\omega)=x$. – Did Aug 12 '11 at 17:20
I'm not sure what you mean by that... are you using $X(\omega)$ as shorthand for doing inverse transform sampling on $X$, with $\omega \sim U(0,1)$? Then I'm left with the same notational need, but for $\omega$. – Sneftel Aug 12 '11 at 17:32
By definition, a random variable is a measurable real-valued function on a probability space. Conventionally $\omega$ denotes an outcome, whence the usual mathematical notation for functions $X(\omega)$ is applicable. This is a great mathematical answer, but in another sense it's not an answer at all, because it just pushes back the question to "how was $\omega$ chosen?" But that realization at least forces you to stipulate whether your question is about mathematics or the physical process of sampling. – whuber Aug 12 '11 at 21:53
@whuber, the question is explicitely purely notational (hence neither, Dieu merci, about the physics of sampling, nor even about mathematics). – Did Aug 13 '11 at 0:32
@Didier Yes, but the notation depends on the context. $X(\omega)=x$ would rarely appear in any applied statistics journals, for instance, but is commonplace in the theoretical ones and in the financial literature. – whuber Aug 13 '11 at 14:26

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