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.

  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 X2 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. x1) or an uppercase number with a number(e.g. X1).

  4. The random variable that is the kth order statistic of n draws from a random variable X is notated as Xkn.

  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 EF(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, Fnew(x)=Fold(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 F2(x) or F(x)2?

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

Answer

  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. X2 is another random variable; whenever X=x, X2=x2.

  3. I would generally use lower cases letters as realizations of random variables. I wouldn’t use X1 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, X1, …, Xn. I would generally write the kth order statistic not as Xkn but as X(k), and note that it is a random variable.

  5. You generally write XF 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 EX where XF.

  7. I would just write Y=max 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.

Attribution
Source : Link , Question Author : OctaviaQ , Answer Author : Karl

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