Why are there many notations for expected value?

Question

I saw from literature that the expected value of a random variable $f(X)$ is either $E f(X)$, $E(f(X))$ or $E[f(X)]$. Is there a standard which one notation should one use? Is the expected value a function $f(X)\to\mathbb R$?

Answer 1

It is totally up to the author. Some authors use a different typeface for the $E$ and they tend to be the ones who avoid parentheses or brackets: $\mathbf{E}X,$ $\mathbb{E}X,$ $\mathsf{E}X$, and so on (including a script E, which I've forgotten how to make).

It is also common to use $\mu$ when there is only one random variable under discussion or $\mu_X$ and $\mu_Y$ when there are several. This make is convenient to write expressions such as $E(X - \mu_X)^2$ without accumulating too many brackets or parentheses. as in $E\{[X - E(X)]^2\}.$

As for your second question: A random variable is a function from the sample space to the real numbers, sometimes written as $\Omega \stackrel{X}{\rightarrow} \mathbb{R}.$ Moreover, if $f$ is a function from the real numbers to the real numbers then $f(X)$ is another random variable. Then we might write $\Omega \stackrel{X}{\rightarrow} \mathbb{R} \stackrel{f}{\rightarrow}\mathbb{R}$ or $\Omega \stackrel{f(X)}{\rightarrow}\mathbb{R}.$ Expectation itself is not a function (because it yields a constant); the word 'operator' is often used for the process of producing that constant.

I see that another Answer by @Karl has appeared with some entirely different notations for expected values. Outside the mainstream of mathematical statistics, there are many more such notations.

One potentially confusing notation for math stat people is common in queueing theory where $L$ can stand for the AVERAGE number of people in a queueing system, $L_Q$ for the average number of people waiting to be served, $W_Q$ for the average time waiting to be served, and so on. The random variables themselves are seldom mentioned and have a variety of notations other than capital letters when they are mentioned.

Adding to the diversity is the routine use of small Greek letters (such as $\xi$ and $\eta$) or small Roman letters (such as $\text{a}$ amd $\text{b}$ for random variables, common in some European and Asian countries.

Answer 2

I've seen $\langle X \rangle$ used as well for expected value. I quite like this as it makes moment generating functions look nice $\langle e^{tX} \rangle$ only has one 'e' compared with $\mathbb{E}\left[e^{tX}\right]$ I guess that's your answer, people choose notation to balance form and function. I suspect there is no standard notation just some are more common than others.