Can a Random Variable be a Sample or just a Single Value?

[manoj]

In my introductory statistics course, I've seen statements like 'Y1 . . . , Yn is a simple random sample without replacement, of size n, from the population.' and that the CLT applies if 'Y1, . . . , Yn are independent and identically distributed random quantities'.
The first suggests that when you say you've 'drawn a random variable from a population', your variable represents a single member of that population, randomly chosen. The second, and other definitions of the CLT or E[X] for example, suggest that a random variable drawn from a population is a random sample sample whose members are randomly chosen.
When someone talks of taking a 'random variable' from a population, is the exact meaning of what the r.v. is flexible, and depends on the context (e.g. on the particular formula) the variable is used in? Or is it more fixed? It's just been confusing me because as I try to understand the meaning of E[X], the CLT etc, I keep getting confused as to what from a statistical population each formula applies to.
Many thanks indeed, really appreciate your help.

 

[fed2]

seems like they are different, in that the first example of

a simple random sample without replacement

, is not 'independant', as is the case with

independent and identically distributed random quantities

Random variables don't have to be independent of one another though, as a matter of necessity.

random variables maps out comes of experiments (which are sets of something you are sampling) to real numbers. That's weird as hell, but important.

 

[gw1]

The first two statements on Y1-n mean the same thing, as I read it. They say 'the standardised residual errors of randomly sampled units would theoretically have a mean = 0' if the sampling units are randomly chosen and enough are chosen to be representative of the population.
The term 'random variable' more often describes the nature of the independent variable (random or fixed) not the individual unit that is randomly selected.

 

[Dason]

I disagree with the statement that "random variable" is more often associated with the independent variables.

 

[gw1]

Happy to reword it, how would you put it

 

[fed2]

is my cat a random variable?

 

[fed2]

it was a trick question, cats can't be random variables, because they are cats.

 

[gw1]

'Cats' could probably be a random variable in a nested model e.g. testing cage size effects on breeding

 

[fed2]

Yes that's true. But wouldn't cats in the same cage be correlated? If cats can be correlated and cats are random variables then random variables can be correlated. That's logic!

 

[Dason]

I mean if we're going to take this seriously then I don't think a cat could be a random variable under the typical definition. It could certainly be a member of a sample space which is the input to a random variable but it wouldn't be a random variable itself.

 

[gw1]

Yes that's true. But wouldn't cats in the same cage be correlated? If cats can be correlated and cats are random variables then random variables can be correlated. That's logic!

I thought one cat per cage

 

[gw1]

I mean if we're going to take this seriously then I don't think a cat could be a random variable under the typical definition. It could certainly be a member of a sample space which is the input to a random variable but it wouldn't be a random variable itself.

Sample space as in randomly chosen from a population? Yes this was the point

 

[fed2]

I mean if we're going to take this seriously

Among the maxims on Lord Naoshige's wall, there was this one: "Matters of great concern should be treated lightly." Master Ittei commented, "Matters of small concern should be treated seriously."

I don't think a cat could be a random variable under the typical definition.

Yes that's a good point. Cats do not themselves know how to map members of a sample space to numbers, although they might be members of the sample space. I think you need human to create such a map, a primate for sure, possibly some intelligent birds like crows, I saw on 'Nature'. Certainly not a cat, at least not mine.

Sample space as in randomly chosen from a population?

Yes, but they themselves are members of the sample space, and do not play any role in assigning numerical values to themselves. That is strictly the purview of the humans or other animals that enumerate and group them.

I thought one cat per cage

Yes sorry, this is the max number of cats that should be placed in a cage unless they are used to one another.