Why is the expected value (mean) of a variable written using square brackets? My question is told in a few words: Why do you write $E[X]$ in square brackets instead of something like $E(X)$? Probably it is not a "function". How would you call it then? This question also applies for $Var[X]$.
 A: I don't. 
You can write in both ways, it doesn't matter. Some don't even use brackets but might instead write $EX$ or $VX$. 
A: I agree with all the above, but also it might has been inherited by the "brackets notation" used to notate discrete functions. For example, a discrete signal is notated as $x[n]$. Just saying...
A: Mathematicians usually adopt square bracket such that $f[g]$, instead of $f(g)$ to indicate $f$ is a functional. Obviously, random variable $X=X(\omega)$ is a function, which makes expectation, variance and so on functionals. But this is only an unestablished convention, not necessary.
A: I too have been in search of this answer recently, "The use of square brackets was used as a strict mathematical definition of what E really was within its context, E[x], with expectation"
This may be due to the notion of polynomial rings, as it stands to reason that expectation fits the properties of a polynomial ring, Perhaps ergo it's seen as an extension of $\mathbb{R}[X]$, hence E[X] - or potentially $\mathbb{E}[X]$?
I am not a mathematician, perhaps someone could confirm this.
