Finally there is the Greek letter Chi to mean the set over which that self-same RV varies. Chi seems deliberately chosen because it resembles "X". That might not be a good idea, but there is a defensible argument for it:
The domain Chi is a piece of mathematical machinery that is logically needed, but is not very important to the main disucssion. He wants to note it down, but also let it fade into the background. It's precesly because you want to pay too much attention to it that you are getting upset.
(I'm more worried that he seems to elide the distinction between the domain of an RV and the underlying sample-space. But then, sample-spaces are also mathematical machinery that aren't needed once the rubber hits the road.)
It might be a little clunky in isolation, but it lets you talk about multiple random variables and their properties in a fairly compact way.
You could write X, codomain(X), and possibleValue(X), but this would be pretty verbose.
The set script-X, the random variable X, and the value x refer to three different but interrelated concepts. Using entirely different letters would likely make it harder to understand.
And the person you're replying to seems to be well aware it is a standard notation: "Can I rant on how bad much mathematical notation is in terms of usability?"
> A discrete random variable X is completely defined by the set of values it can take, X , which we assume to be a finite set, and its probability distribution {pX(x)}x∈X . The value pX(x) is the probability that the random variable X takes the value x.
Actually, we don't have that, because copy/pasting it here loses essential information. This paragraph uses "X" to refer to three entirely different concepts, distinguished only by the font used.
Not only that, but it relies on subscripting and case, yet uses letters like "p" and "x" which difference in case is mainly in their vertical size and position.
Alright, rant over.