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I sometimes get confused when reading statistical definitions when they mention random variables (RV). Are they talking about a single draw? Are they talking about an estimator? To make things clear for myself (and others reading the question) I have created a list of statements that I think are true. Is there anything that I have said here that is false?

1) An element in a population IS NOT a RV since it cannot take on a set of different values.

2) A parameter of a population (e.g. mean) IS NOT a RV since it is based on elements which are not RVs.

3) A random draw from a population IS a RV since its realization is based on the proportion of that element in the population.

4) A sample is made of RVs since each element of the sample is itself a RV.

5) A statistic of a sample (e.g. sample mean) IS a RV since the sample is made up of RVs.

6) An estimator of a population parameter (e.g. sample mean) IS a RV since it is based on a sample and the sample is made up of RVs.

7) A sequence of statistics (e.g. sequence of sample means) is a sequence of RVs.

8) A sequence may converge to either to a constant or a RV.

EconStats
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    [What is meant by a random variable?](http://stats.stackexchange.com/q/50/919) addresses most of these points. In (1) it is not apparent what is meant by "take on". (3) is also a little unclear and could provoke some objections based on distinguishing physical from mathematical concepts. (8) might also cause some trouble for people to discuss because nothing guarantees the convergence of arbitrary sequences of random variables; when such a sequence converges, it converges to a random variable--which could be a *constant* random variable. – whuber May 29 '14 at 03:00
  • @Whuber Thanks for the link, that is definietly a v. detailed explanation, I'll make sure to read that. With regards to 1) I was trying to get across the following point. If our population was the ticket box and we only had two tickets, R and D, then the ticket R is always equal to R? With regards to 8) I was trying to distinguish between convergence in P and convergence AS. Convergence in P implies convergence to a constant while convergence AS implies convergence to a constant RV. It was actually looking up a lot of LLN stuff that got me interested in this question! – EconStats May 29 '14 at 10:55
  • @Whuber Unfortunately your comment on 3) is way above my intellectual pay grade – EconStats May 29 '14 at 10:58
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    (The different kinds of convergence are not distinguished by what the sequence can converge to.) In the tickets-in-a-box model, the tickets never change, nor do their proportions in the box. A random variable is a consistent way to write numbers on the tickets. That all there is to it: that's the mathematical model which is formalized in the language of sigma-algebras, probability measures, and measurable functions. It can be applied to the real world only after we have shown (or assumed) that our data *behave as if* they had come from drawing tickets from a box. – whuber May 29 '14 at 15:51

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