Notation conventions for random variables and their distributions

Question

I get confused on the proper notations of meanings, as well as the meanings of some notations relating to random variables and their distributions. Below, I will list things that I think are true, as well as things that I don't understand, and I would love input/corrections. I have labeled each point/question with a number for ease of reference. If it is not appropriate to list items in a single question like this, please let me know. I thought it would be ok since they are all short.

1. A random variable is notated by a capital letter, e.g. $X$.

2. What does an operation on a random variable mean? (e.g., how do you interpret $X^2$ in words?).

3. A specific draw from a random variable is notated by either the lowercase letter (e.g. $x$) or the lowercase letter with a subscript (e.g. $x_1$) or an uppercase number with a number(e.g. $X_1$).

4. The random variable that is the $kth$ order statistic of $n$ draws from a random variable $X$ is notated as $X_{kn}$.

5. Is there a shorthand way to write "X is the random variable that is distributed by F(x) (or "cdf F(x)" or "B(a,b)" or any way to characterize a distribution)"?

6. Can I write $\mathbb{E}F(x)$ to mean the expectation of the variable distributed according to $F(x)$?

7. If I perform an operation on a variable X's cdf, for example, $F_{new}(x) = F_{old}(x)^2$ to get the cdf of the maximum of 2 draws from $X$, can I notate that in terms of $X$ somehow?

8. Is the appropriate way to write $(F(x))^2$ succinctly $F^2(x)$ or $F(x)^2$?

9. Is there any notational difference between a discrete and a continuous variable?

Answer 1

1. I like to say: a random variable assigns a number to each possible outcome of a random "experiment", where a random experiment is some well-defined process with an uncertain outcome.

2. $X^2$ is another random variable; whenever $X = x$, $X^2 = x^2$.

3. I would generally use lower cases letters as realizations of random variables. I wouldn't use $X_1$ this way; it would be another random variable.

4. I wouldn't talk about $n$ draws from a random variable. I would talk about $n$ draws from a distribution, which would give $n$ independent and identically distributed random variables, $X_1$, ..., $X_n$. I would generally write the $k$th order statistic not as $X_{kn}$ but as $X_{(k)}$, and note that it is a random variable.

5. You generally write $X \sim F$ to say $X$ is a random variable with distribution $F$.

6. I've never seen that notation for the mean of a distribution. I'd say $\mathbb{E} X$ where $X \sim F$.

7. I would just write $Y = \max(X_1, X_2)$ where $X_i \sim \text{iid } F$.

8. I guess either might be understood, but probably $[F(x)]^2$ is most clear, and while it's more cumbersome to type, it doesn't really take up much more space.

9. There's not generally a notation difference between discrete and continuous variables, except that you generally wouldn't choose $N$ to be a continuous random variable.