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Suppose I have a variable $x\in\mathbb{D}_{X}$, with $\mathbb{D}_{X}$ being its domain. What is the mathematical correct way of expressing "$\mathbb{D}_{X}$ is discrete" (i.e. not continuous)?

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    $\begingroup$ Discrete variable does not have to be non-negative. Moreover, discreteness is about countable number of categories, not about being represented by integers. $\endgroup$
    – Tim
    May 30, 2018 at 12:20
  • $\begingroup$ thanks for the comment, I removed my obviously wrong guess $\endgroup$
    – Boern
    May 30, 2018 at 12:23

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The exact definition of a discrete random variable is quite subtle, and is discussed in detail in this question. To cut a long story short, a countable set of possible outcomes is a sufficient but not necessary condition for a random variable to be discrete. More strictly, a random variable is discrete if it has probability one of falling into a countable set of possible values (either finite or denumerable). Also, random variables are not just numbers in a set - they are functions from a sample space to an outcome set. Hence, it is a slight abuse of notation to refer to random variables as being values in an outcome set.

Suppose we set aside this complication, and consider the simple case of using a countable set of possible outcomes to define a discrete random variable (which is sufficient). Without any further specification of the possible outcomes, those possible outcomes could be anything; they could be integers, or rationals, or just some arbitrary numbers. There is no inherent problem with choosing a new symbol to represent your countable set, though it is usual to use a script-font with the same letter as the random variable. Since all countable sets exist in a one-to-one relationship with simple sets of integers, it is also usual to denote countable sets by reference to the integers. (In some cases it might be simpler to denote the set by reference to the rationals, or some other countable set, but it will depend on the structure of your set.) For simple set structures, here are some common notational conventions.

Standard notation conventions for finite sets: For a finite set with $n \in \mathbb{N}$ elements, it is usual to denote the set as $\mathscr{X} = \{ x_1, ..., x_n \}$ and then refer to outcomes as $X=x_i$ for some $i \in \mathbb{N}_n$.

Standard notation conventions for bounded denumerable sets: For a denumerable set with a lower bound one the outcomes (i.e., a smallest value), it is usual to denote the set in a way that lists the values in increasing order, as $\mathscr{X} = \{ x_1, x_2, x_3, ... \}$ where $x_1 < x_2 < x_3 < ... $ and then refer to outcomes as $X=x_i$ for some $i \in \mathbb{N}$. If the set has an upper bound instead then this notation can be reversed to list elements in decreasing order.

Standard notation conventions for unbounded denumerable sets: For an unbounded denumerable set, it is usual to denote the set in a way that lists the values in increasing order, as $\mathscr{X} = \{ ... , x_0, x_1, x_2, ... \}$ where $... < x_0 < x_1 < x_2 < ... $ and then refer to outcomes as $X=x_i$ for some $i \in \mathbb{Z}$.

The above notational conventions are common when analysts deal with discrete random variables. Discrete sets are usually denoted by direct analogy to the natural numbers, or integers, or a subset of these. In cases where the discrete set of interest has a structural analogy to some other set (e.g., the rational numbers) then this analogy might be preferred. If the outcomes are just a string of consecutive integers, you can simplify accordingly.

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  • $\begingroup$ Some recent questions about discrete random variables have uncovered the importance of emphasizing (a) the distinction between this meaning of "discrete" and that of a discrete topological space and (b) the important role played by the underlying probability. This is brought to the fore when one considers discrete random variables with uncountable sets of values, but where most of the values have no probability of occurring. It might also be worth remarking that the notation in the question, "$x\in\mathbb{D}_X,$" although mathematically meaningful, does not mean what the OP intends. $\endgroup$
    – whuber
    Jun 18, 2018 at 2:21
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    $\begingroup$ @whuber: Thanks for that - I initially interpreted this question as a purely notational one, but you're right that he is also asking about the meaning of "discrete" random variables. I have added an opening paragraph to be more accurate on this point, with a link to a previous answer on the substantial question. I have kept the later notational discussion, since that also bears on the question. $\endgroup$
    – Ben
    Jun 18, 2018 at 3:18

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