> 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.
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.
Technically speaking, there are other cultural spaces than US/English. In one such space E[X] is written M(X) and called 'average value'. But that's quibbling. 'Expected value' is simply 'weighted sum' over possible values with respective probabilities as weights: weighted average value. Not exactly rocket science if one groks what a probability distribution is. But even that is quibbling. The more interesting point is that some would rather starting learning from concrete applications instead of pacing through a seemingly endless dry litany of definitions. Cryptic notation is unhelpful for this style of learning.
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.