Mathematical Notation Is Awful

>Two, wikipedia refers to E(X) and E(Y) as the means, not the expected value. This gets even more confusing because, at the beginning of the Wikipedia article, it used brackets (E[X]), and now it's using parenthesis (E(X)). Is that the same value? Is it something completely different? Calling it the mean would be confusing because the average of a given data set isn't necessarily the same as finding what the average expected value of a probability distribution is, which is why we call it the expected value. But naturally, I quickly discovered that yes, the mean and the average and the expected value are all exactly the same thing! Also, I still don't know why Wikipedia suddenly switched to E(X) instead of E[X] because it stills means the exact same goddamn thing.

IIRC, the mean is the particular case of the expected value where the transformation function is the identity function.

The general expression of the expected value, where f(x) is the probability density function and g(x) is the transformation function, is:

E[g(X)] = \int{g(x) * f(x) * dx}

Let g be the identity function (g(x) = x), then E[g(X)] is the mean.

For example, let's say that you bet on a throw of dice such than you win 2$ if the result is odd, lose 3$ if the result is 2, and win or lose nothing otherwise. Here, the event X is the throw of dice: you have 6 possible outcomes. Your transformation function is the gain you get for each outcome, therefore g(x) is defined by:

1 -> +2

2 -> -3

3 -> +2

4 -> 0

5 -> +2

6 -> 0

Then, the expected value E[g(X)] can be calculated by summing all the g(x) multiplied by the probability p(x) for all x (all outcomes).

The first point of the article, in my opinion, is more about statistics than math in general, and I believe people find it weird because of the confusion between X and x, the former being a random variable associated with a distribution function, the ater being a value in the domain of X.

The second point of the article is just ease of notation. The derivative of a function doesn't exist; a derivative is always made with respect to a variable (df/dx), but if a function has only one variable, then the derivative of the function is defined as the derivative of the function with respect to the one and only variable and we don't need to write this only variable (f'(x)).

You could always write the variables of a function (df(x, y, z)\dy instead of df/dy), but that's just a waste of time, specially when you have to write it for every line of the demonstration.

the problems I had were mostly with good names. So names that are 1. already used in everyday life AND 2. really fit the concept.

These words have definitions in my head that I use often and now math uses them differently.

Like 'Menge' (german for Set) which has 'almost' the same definition in math and everyday use.

I like the physics approach more, gluons, quarks, protons, etc. these are all words that never had any definition in my life in the first place

Leibniz notation is very interesting as it is not technically correct but it is still is used in lots of maths today. Recommended: http://math.stackexchange.com/questions/21199/is-frac-textrm...

This a thousand times. I boycott math notation. My most memorable math moments are:

- giving a talk on notation in high school (I was not tasked to do this, I decided to do it on my own) because I was FREAKED OUT by how we were using tons of symbols nobody had ever explained or defined

- converting all Math I encountered in University to Common Lisp programs to get rid of the shit notation

- bursting into crazy laughter when after five algebra lectures the prof notices that the students parse his notation differently than he does

Give it names, use S-Expressions.

It's good practice anyway in LaTeX to define your own aliases and use them for semantic information. Like in one of my recent projects, I define:

    \newcommand{\disjointunion}{\sqcup}