Notation confusion in the Wikipedia article on the Law of Large Numbers

Question

In my infamous attempt at mastering (at my humble level) the "art" of probability and statistical theory, I was reading the Wikipedia article on the Law of Large Number and got confused by a couple of notations.

1 - mean vs expected value

First it is written:

$\bar X \rightarrow \mu$ for $n \rightarrow \infty$

For me $\mu$ is the "population" mean and not the expected value. The definition of the law is that the sample mean converges to the random variable expected value as the sample size approaches infinity (not its population mean). I realize the population mean and the expected value are equal but they are not computed the same way and I found this notation misleading (because it doesn't follow directly the definition). What do you think?

2 - random variable vs observation

Second it is written:

$\bar X = {1 \over n}(X_1 + X_2 + ... + X_n)$

"where X1, X2, ... is an infinite sequence of i.i.d. integrable random variables with expected value $E(X1) = E(X2) = ...= \mu$."

Why I am confused is that for me the sample mean is computed as the average of n observation where the observations are produced by a random variable X. So it would be for me at least less misleading to use x lower case (observation, realization) instead of X uppercase. Or is correct here to use X and if so why?

EDIT: I understand you need to write $E[X_1]$ and can't write $E[x_1]$. The expected value of an observation wouldn't make much sense. But that what's the meaning really of $X_n$ in $\bar X = {1 \over n}(X_1 + X_2 + ... + X_n)$. For me a random variable is a function not really a number? An average of functions?

But I am not an expert so there might be an explanation to both notations? Thank you.

Answer 1

First, in statistics, "mean value" and "expected value" are synonyms: see here.

Second, the notations in mathematics are quite flexible (there're some standard notations, of course; real numbers are always $\Bbb R$, for example) in terms of how you call your variables. You can write them in uppercase, lowercase, greek or latin alphabet, sometimes we use hebrew and cyrillic symbols (might have missed something, I suppose). You are free to introduce your own conventions of notations, and they don't necessarily coincide with the notations on wiki. As a personal example, in Russia it's quite common to note Lebesgue spaces as $L_p$, while in international literature the standard notation is $L^p$.

In your particular example, you note your variables in uppercase latin, so in my opinion it's logical to note the result of arithmetic operations over them in the same uppercase latin, to reinforce that this result retains the same nature as the terms in the sum. To illustrate, one of common notations for linear algebra uses lowercase latin for vectors and lowercase greek for scalars to underline that these two entities have different nature.

Hope this helps.