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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 $Var[X]$.

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4 Answers 4

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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.

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  • $\begingroup$ What do you mean by saying "$f$ is a functional" when also stating that expectation and variance are functionals? It sounds to me like: "Yes, $f[g]$ shall not be confused with $f(g)$ but essentially it is the same." $\endgroup$
    – Xiphias
    Nov 19, 2013 at 13:41
  • $\begingroup$ @Tobias $f$ is to $g$ what $E$ is to $X$. And yes, both () and [] are of the same meaning with different styles. $\endgroup$
    – Shuchang
    Nov 19, 2013 at 13:51
  • $\begingroup$ "Mathematicians usually adopt square bracket such that f[g], instead of f(g) to indicate f is a functional." Any reference for this? $\endgroup$
    – Did
    May 15, 2014 at 18:34
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    $\begingroup$ @Did Probably nope. We write $f[g(x)]$ instead of $f(g(x))$ for a good writing style. And usually we don't care about $x$ in a functional so just ignore it together with (). This leaves [] there. $\endgroup$
    – Shuchang
    May 16, 2014 at 3:15
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    $\begingroup$ To be frank, I do not understand why you present the use of brackets as a general or standard practice. It is not. "Mathematicians" and "usually" in your answer are both misleading. "We" in your comment is misleading as well--unless you explain who are this "we". (Note that "unestablished convention" is an oxymoron.) @upvoters Care to explain your upvote? $\endgroup$
    – Did
    May 16, 2014 at 5:29
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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 $VX$.

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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...

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  • $\begingroup$ In my experience engineers write $x[n]$, mathematicians write $x_n$. $\endgroup$
    – mathandy
    Feb 14, 2018 at 9:28
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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.

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  • $\begingroup$ You should at least summarize the contents of the page linked to, it can go away at any moment. $\endgroup$
    – vonbrand
    May 15, 2014 at 18:55
  • $\begingroup$ Ok thanks vonbrand, I've edited to include reasoning with links to wikipedia - which hopefully shouldn't disappear $\endgroup$
    – Tjad Clark
    May 15, 2014 at 19:45
  • $\begingroup$ The interpretation of an extension of the polynomial ring notation is pure fantasy. $\endgroup$
    – Did
    May 16, 2014 at 5:22
  • $\begingroup$ Thanks Did, could you perhaps elaborate why Expectation is not a polynomial ring ? $\endgroup$
    – Tjad Clark
    May 18, 2014 at 13:28
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    $\begingroup$ lol, and as a matter of fact, the link did go away! thanks for the edit! :) $\endgroup$
    – Harsh
    Dec 19, 2021 at 21:27

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