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I've just started statistics (rather late) as the part of my course, I have done a little bit of logic where notion of variable is well defined.

Typically a 'variable' is treated just as a symbol usually one for which we look at different 'values' (assignments in the context of formal languages)

Sometimes 'random variables' are treated as if they work in a similar way, they are discussed as 'taking values' and sometimes we will discuss the probability of the random variable X taking a value that is an element of a set S as $P(X∈S)$ I could see this as for a symbol X the associated probability for the assignment that makes the fomula true.

This where I feel I am missing something, if $X$ is a function or mapping, then it would be $X$ at some value that is an element of S and not $X$ itself, if 'X' is a variable that varies over the values of a function and not denoting a function itself then that would be fine, but why are Random variables treated like this when defined as functions?

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    $\begingroup$ A random variable is a function that maps random events to (numerical) values. The range of this function is the set of possible values, which can be discrete or continuous, let $\mathbb S$. In the expression $P(X\in S)$, $S$ denotes a subset of $\mathbb S$. One studies different realizations of the variable, corresponding to different events. $\endgroup$
    – user1020730
    Feb 23, 2023 at 16:19
  • $\begingroup$ In the context of it as a function surely $X∈S$ is misleading as it would suggest, as $X$ is a function, that $S$ is a set of functions? $\endgroup$
    – Confused
    Feb 24, 2023 at 11:01
  • $\begingroup$ Nah, the argument is omitted. $\endgroup$
    – user1020730
    Feb 24, 2023 at 11:02
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    $\begingroup$ I guess it is easier then in that case, I'll keep that in mind that its really $X(w)∈S$ and the associated probability of $w$ being such that $X(w)∈S$ $\endgroup$
    – Confused
    Feb 24, 2023 at 11:07

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