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Let $X$ be a discrete random variable that can take the values $1, 2, \textrm{and}\ 3$. What is the conventional way to write this mathematically? Is it just $X\in\{1,2,3\}$ or should I somehow write the pmf.

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  • $\begingroup$ If you only need to write the possible values taken by the random variables, your notation would seem to do that. The pmf specifies the probability that the various values occur, which is a different piece of information than you ask about. $\endgroup$
    – Glen_b
    Commented Sep 21, 2015 at 23:51

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There are sloppy ways and rigorous ways. The sloppy ways are shorthands, like "$X\in\{1,2,3\}$", that are either nonsensical or (in this example) just plain wrong when interpreted according to the correct conventional meanings of the symbols. (The second statement literally means $X$ is one of three specified integers--which aren't random variables at all.) Such a shorthand can be effective in contexts where (a) its meaning is defined and (b) set-theoretic notation will not otherwise be used.

Recalling that all random variables are measurable functions defined on probability spaces, a standard way to stipulate that $X$ can take on a given set of values is to use functional notation to specify its image, as in

$$X:\Omega\to\{1,2,3\}$$

or

$$X(\Omega) = \{1,2,3\}.$$

Many statistical writers eschew such notation because they prefer to suppress all references to $\Omega$, which is intended to remain abstract, or they even think of random variables as being some kind of class of objects in which $\Omega$ is not even a definite set. An equivalent notation avoids referencing $\Omega$, such as stipulating

$$\text{Image}(X) = \{1,2,3\}.$$

Very confusingly, some statistical writers use the term "domain" to refer to the image of $X$ (whereas in mathematics the word "domain" invariably refers to $\Omega$!). A Google search will easily turn up such uses of "domain." Others use phrases like "defined on" or "take on," such as "$X$ takes on the values $\{1,2,3\}$" or "$X$ is defined on $\{1,2,3\}$" (by which they really mean the probability mass function of $X$ rather than $X$ itself).

There are indirect ways to refer to the image of $X$. For instance, real-valued random variables are often thought of as being almost interchangeable with their distribution functions. The support of such a function has a well-established definition in probability and measure theory; in the case of a finite discrete random variable $X$, it will coincide with the image of $X$. People who write about such things typically adopt some mnemonic notation for this, such as

$$\text{supp}(X) = \{1,2,3\}.$$

Finally, this helps us understand how such confusion about the meaning of "domain" can arise. If we were to conflate the random variable $X$ with its probability mass function (pmf) $p_X$, given by the probabilities

$$p_X(x) = \Pr(X=x),$$

then in the example $p_X(x)\ne 0$ only when $x \in \{1,2,3\}$. We could, if we wished, restrict $p_X$ (which notionally is a function defined on $\mathbb{R}$) to the subset $\{1,2,3\}\subset\mathbb{R}$ without losing any of the information it conveys. This would make $\{1,2,3\}$ the (mathematically correct) domain of the restricted $p_X$.

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A full description of $X$ would include the pmf, yes. For example ... $$ X \in \{1,2,3\} \\ \mathbb{P}(X=1)=\frac{1}{12} \\ \mathbb{P}(X=2)=\frac{7}{12} \\ \mathbb{P}(X=2)=\frac{1}{3} $$ ... which looks a bit clumsy, or do it in words: $X$ is a discrete random variable in $\{1,2,3\}$ with probabilities $p_1=\frac{1}{12}, p_2=\frac{7}{12}, p_3=\frac{1}{3}$ respectively.

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