n observations from a random variable VS. 1 observation from n i.i.d random variables

Question

I have a question about one single random variable vs. a bunch of random variables:

If there are n observations from a single RV (call it $X$), and there are n observations in total from n i.i.d RVs (call them $Y_1,Y_2,...,Y_n$), 1 observation from each of the $n$ RVs. The distribution of $Y_n$ is the same as $X$.

Do the two groups of observations have the same distribution?

(If strictly speaking, two groups of observations cannot have distribution because they are not RVs and only RVs can have distribution. So the question is do the two groups of observation have the same mean and variance (or any other statistical parameters) when $n\to+\infty$)

Maybe due to my understanding of RV is not deep enough, I could not find the correct direction to solve the problem. Neither do I know if this question is meaningful or asked correctly.

Answer 1

When modelling a sample $(x_1,\ldots,x_n)$ as an $i.i.d$ sample from a given distribution $F$, the correct way of modelling is to see this sample as the realisation of $n$ random variables $(X_1,\ldots,X_n)$ made of $n$ independent random variables identically distributed from $F$:

$$(x_1,\ldots,x_n)=(X_1,\ldots,X_n)(\omega)\qquad\omega\in\Omega$$

The concept of $n$ realizations of a single random variable is a shortcut that is not well-defined because one cannot handle independence with a single random variable.

Answer 2

I think it depends on the theoretical and mainstream approach. This approach could not be proved, but it is such an axiom. I looked in my books on math. stat., authors do not mention this question.

Some points about that:

1. Math. stat. theory approach based on tests and theory derived from i.i.d. condition. With this condition and assumption that the random sample contains n observations for n random variables we know how to get unbiased, consistent and effective estimates. In case when we assign n observation got from one rv, we cannot apply CLT, MLE, F, chi square etc.

2. As mentioned above, it is hard to incorporate independence in statistical approach if we consider all n observations from one random variable. For example, in average, temperature rises in the summer, i.e. previous observation could influence the next level of temperature. I.e. cumulative distribution could change through the time. If we have only one rv, we could not explain this influence. Also, there are some “structural breaks” in time series or non time sample, this means that we could have different distributions.

3. Remind definition:

For a given sample space S of some experiment, a random variable (rv) is any rule that associates a number with each outcome in S. In mathematical language, a random variable is a function whose domain is the sample space and whose range is the set of real numbers

Could you, please, prove that the set of real numbers does not change through the time or does not depend on smth? But in the case when there are only 1 observation for each rv this question drops.