Notation conventions for random variables and their distributions 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. 


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

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

*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. $X_1$).

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

*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)"?

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

*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?

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

*Is there any notational difference between a discrete and a continuous variable?
 A: *

*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.

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

*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.

*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.

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

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

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

*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.

*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.
