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In a recent post of mine Confusion over the definition of "source code" in information theory some commenters pointed out that the range of a random variable need not be subsets of $\mathbb{R}$.

Strictly speaking, I kind of agree with them.

However, I could not find any type of application for random variables with range such as

$$\text{Range}(X) = \{black, white\}$$

Can someone point to a resource or examples where these random variables would be useful?

Sin Nombre
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    Random matrices, random graphs, stochastic processes, random fields, random measures, ... . The list is endless. – Stefan Perko Oct 11 '22 at 14:27
  • A coin toss has also such a range (H or T) – Peter Oct 11 '22 at 14:27
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    Although, obviously the coin toss example is included by identifying $H = 1$ and $T = 0$. – Stefan Perko Oct 11 '22 at 14:29
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    I’m voting to close this question because it should rather be a comment to recent post of OP that OP refers to. – Kurt G. Oct 11 '22 at 14:30
  • @KurtG. I think this is a question of its own, ten years ago or so when I was learning probability theory, no one ever told me that "randomness" and "stochasticity" where the same and that "random matrices", "stochastic processes", etc. were all part of a single definition. It took me several years and reading a few (French books) to realise what was obvious all along. I am unsure why several books love to split the definition so much so that only ever causes confusion (e.g. my own and OP's). – William M. Oct 11 '22 at 14:59
  • Yes, often random and stochastic are the same but not always the case. For instance, a stochastic differential equation and a random differential equation are not the same thing. – oliverjones Oct 11 '22 at 15:08
  • @WilliamM. I don't see how confusion between "random" and "stochastic" is relevant here. $$\quad$$ This is a bad question, which can provoke good answers, but it is still a bad question. You could further develop your own answer by touching upon Bochner integral and Dirichlet process, thinking about measurable spaces not representable as reals with Borel or discrete measure... but unfortunately this question is about coin tosses and bit encoding. :-/ – paperskilltrees Oct 11 '22 at 15:14
  • @paperskilltrees I agree. – Peter Oct 11 '22 at 16:36
  • @WilliamM. No problem. I just had the quick impression that the question was almost answered by the links in the comments of OP's recent post. – Kurt G. Oct 11 '22 at 16:52

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Suppose we both draw a card at random from a deck of cards. Let $X$ be the r.v. representing my card and $Y$ the r.v. representing yours. Now we can ask questions like "are $X$ and $Y$ independent?". We can define a new random variable $Z=s(X)$ where $s$ gives the suit of a card, and so on. The random objects we're working with are not necessarily numeric quantities.

Karl
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  • Thanks for this example. But in your example $Z = s(X)$, why would we go through the trouble of defining $X$ when we can directly define $Z$ as the suit of a card? – Sin Nombre Oct 11 '22 at 18:24
  • You certainly could do that instead. But e.g. we might also care about the rank $r(X)$ of the drawn card and want to study relationships among these variables. For example, are $s(X)$ and $r(X)$ conditionally independent given that $X$ is not the ace of spades? – Karl Oct 11 '22 at 20:13
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Your question is a matter of language and of taste for definitions.

Usually people talk of "randomness" or "stochasticity" when the underlying measure space is of positive measure and of total mass equal to 1. Any mathematical object defined on that space will be called "random something" or "stochastic something." Generically, one uses "random object" or "random element" for this all-encompassing definition. So, if $(X, \mathscr{X}, \mu)$ is a probability space and $f:X \to Y$ is a measurable function ($(Y, \mathscr{Y})$ a measurable space), then $f,$ by pure definition/language is called a "random object" or "random element." Stochastic processes, random variables, matrices, etc. all fall into this category.

When $Y = \mathbf{R},$ one uses "random varible"; when $Y = \mathbf{C},$ "complex random variable"; when $Y = [-\infty, \infty],$ "extended random variable"; when $Y = \mathbf{R}^d,$ "random vector"; when $Y = \mathbf{C}^d,$ "complex random vector"; when $Y = \mathbf{M}_{(m,n)}$ (matrices of dimensions $(m,n)$), "random matrix", you can add "real" or "complex" should you require to; when $Y = E^\mathbf{N}$ ($(E, \mathscr{E})$ another measurable space), "discrete time stochastic process"; and so on.

William M.
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