I really cannot conceive how one can learn what the concept of “expected value” of a “random variable” means, without encountering E[X] notation. This is a technical concept having a technical meaning, and any place that actually defines this meaning will teach you this notation. If you see this notation for the first time in some academic paper, but haven’t ever read any probability textbook, it means that you almost never actually learned the concept, which is my entire point. You might have some intuitive understanding derived purely from the literal meaning of the words “expected value”, but without actually getting technical, this intuitive understanding is mostly superficial, and, as such, not very useful. You won’t be, for example, be able to answer such fundamentally important questions like “is expected value of sum of random variables a sum of expected values of each? Is expected value of product a product of expectations?”. You can’t know answers to these questions without having ever seen E[X] notation, and if you don’t know the answers, your problem is with the actual concept, not notation.
> PS. I agree that the bigger obstacle is the lack of proper big picture 'why do we even bother with these concepts / theorems'. At an extreme, there is (used to be?) a certain style of math books consisting exclusively of a dry litany of 'Definition 1.2.3', 'Theorem 3.2.4', 'Corollary 2.3.1'. Very rigorous and very difficult to ascertain what problems they were trying to solve.
I agree that it very much often is a problem. It’s not a problem of notation, though.
Sorry, but I didn't learn E[X] notation. I learned E{X} notation. Notations aren't even consistent across the lectures I have attended.
Sometimes notation elides 'obvious details'. The details are obvious to those that have already gone through the learning curve, which is easier if one is strongly connected to other working mathematicians than for outsiders. Mathematicians barely notice inadequate or unusual notation. Outsiders struggle and spend significant energy just deciphering the notation.
Anecdote 1: Sometimes notation is too terse: single letters. Granted, efficient for whiteboard scribbling. Would be really nice to standardize an appendix for notation. E[X] = <expr>. Hmmm, what could E be? By the fifth paper, somebody bothers to write 'expected value' in plain English and the mystery in unambiguously clarified. In a voice-based interaction this is a non-issue, not so for those that only have text to deal with. This compounds as a novice has to juggle a set of mysterious symbols with tens of elements.
Anecdote 2: Long long time ago (before ubiquitous email or http:// took off) a young student spent some time working through a handful of type theory academic papers that somehow trickled into his corner of the Universe. The type inference rule notation (https://en.wikipedia.org/wiki/Type_rule), which is rather straightforward in retrospect, ate up more time that he's willing to admit. Γ is just a set of judgments and Γ |- expr is just a notation for 'Γ includes the judgement expr'? Then why don't they use the standard expr ∈ Γ, this is confusing? Are there some examples? Is there some code that one could possibly run through a debugger? The questions remained unanswered, as there was noone he knew in the same linguistic sphere interested in the topic. And the papers themselves never detailed such obvious notation details.
PS. I agree that the bigger obstacle is the lack of proper big picture 'why do we even bother with these concepts / theorems'. At an extreme, there is (used to be?) a certain style of math books consisting exclusively of a dry litany of 'Definition 1.2.3', 'Theorem 3.2.4', 'Corollary 2.3.1'. Very rigorous and very difficult to ascertain what problems they were trying to solve.