Is there a concise way to mathematically write the following statement:
$\zeta$ is a randomly sampled value from a cumulative distribution function $F_X$ of a random variable $X$.
$\zeta~=~?$
More precisely, I want to have a function $\Psi$ that associates each element from a set $S$ with a randomly sampled value from a given CDF $F_X$.
$\Psi : S~\to~?$
Thanks in advance for your answer(s).
NS
You would say $X_1, X_2,\ldots$ are independent and identically distributed samples from the distribution whose CDF is $F_X.$ It is ok for this sequence to be infinite. If $\xi$ is one of these samples and you want to write $\xi=?,$ you need to say which one and write $\xi =X_i.$ If you want each element of the set $S$ to be associated with one of the $X_i,$ (presumably uniquely) then let $A =\{X_1,X_2,\ldots\}$ and consider one to one functions $S\to A.$
The issue is that “a random sample” is underspecified, at least it is when there are multiple random samples under consideration. But we have a mathematically precise way of defining a sequence of iid samples, and we can proceed to treat the random variables in that sequence as ordinary mathematical objects.
If there is a name for the distribution, like $N(0,1),$ it is common to abbreviate like $$X\sim N(0,1)$$ for a single sample or $$ X_i \sim_{i.i.d.} N(0,1)$$ for an independent identically distributed sequence. I wouldn’t consider it too grave an abuse of notation to substitute $F_X$ for $N(0,1),$ effectively naming the distribution by its CDF.