# Confusion between definitions of random variable (especially sequence of random variables) and random sample

I am having a hard time understanding the actual definition of random variable (especially sequence of random variables) and a random sample, as well as the relation between these definitions. Unfortunately this confusion makes it hard for me to understand some of the concepts in statistics.

A random variable is sometimes called a "variable" which value varies according to a probability distribution. At other times, it is called a function that maps from the probability space to the number space. This is confusing already, but I am to somehow couple these two definitions (I guess each of these is more appropriate depending on the context).

Which confuses me more is "sequence of (most frequently i.i.d.) random variables" and "a sample coming from a random variable". I mean, I know the "random variable" term is there to emphasize the fact that we are not talking about any particular values, but actually "a sample coming from a random variable" seems just the same for me as long as we don't specify any particular values.

Is there any practical difference between those two? Or can they be used interchangeably depending on what you want to emphasize? Especially, what is the difference between a sequence of independent and identically distributed random variables and a sample of independent and identically distributed observations taken from a random variable?

I can see the first one in the definiton of CLT and the other in definition of MLE (not exactly how I specified it, but I believe the sense is the same.

Rephrasing the question in programming way, if in R i run the following code:

values <- rnorm(100)


Is this a sample of size 100 taken from one random variable, or is this a realisation of 100 identically distributed random variables? That's the other problem, it seems that a random variable is sometimes treated as one value, and at other times it is treated as a generator from which you can sample.

I am confused and would appreciate any hints or good sources to understand this properly.

• This doesn't answer your question, but when you add normally distributed variables, the answer is also normally distributed. I'm not sure there's a good answer to this question, but feel free to contact me and I might be able to help. – barrycarter Sep 1 '16 at 17:59
• Hey, thanks for the answer. Yeah I know it works for addition as well (as the average is basically a sum divided by a constant). I was thinking quite a lot about my question today, and the closest I can get with my intuitive understanding is the following: when you talk about a sequence of random variables, you wish (or have to) emphasize that there are many possible experiments and you are not talking about any of those in particular, neither you have to explicitly say that "something has certain attributes after performing many experiments". When it comes to a random sample though, what... – Matek Sep 1 '16 at 20:23
• ... you are saying that there is a one experiment you need to perform and you focus your description on that one particular experiment/sample, not emphasizing on many possible experiments. Something like that. Additionally, when you want to describe something in the area of random variables, you need to actually keep the random variables in your talk, like in CLT. It would be strange to say "an average of a sample is a random variable" - "an average of a sequence of random variables is a random variable" seems more meaningful. – Matek Sep 1 '16 at 20:31
• This answers (part of) your Q: stats.stackexchange.com/questions/50/… – kjetil b halvorsen May 23 '18 at 18:02