Central Limit Theorem - question terminology?

Question

I've never done stats before and I'm not sure I'm correctly understanding the wording of this question:

Let the sequence of n random variables {$X_1, X_2, . . . , X_n$} be independently obtained from some probability density function pX(x).

If $E[X_i] = μ$ and $Var(X_i) = σ^2$ are finite then as n increases to infinity, according to the Central Limit Theorem the random variable $ S = \sqrt{n}(\bar X_n − μ)$ approaches which distribution?

Justify its type (name), and hence compute its mean and variance.

How can S be described as a 'random variable'? The question seems to describe a single sample set, rather than multiple sample sets - am I misunderstanding the terminology here? If it is a single sample set then how can S approach a distribution?

I thought I understood the central limit theorem concept. I'd make a joke about the variance between reality and my understanding but I'll spare you that....

thanks

Answer 1

I appreciate the joke. $\bar{X_n}$ is a "random" variable since it is the sum of "random" quantities. It is not the empirical mean of any one particular sample, it is the variable representing the empirical mean of all possible samples, essentially.

For example, if $X_1$ and $X_2$ are independent standard normal variables, $\bar{X_2}=\frac{1}{2}(X_1+X_2)$ is again a random variable normally distributed with mean zero and variance $1/2$. The distribution tells you the probability of observing a mean value in a certain range if you took any sample of independent normal variables, and is no more or less random than $X_1$ and $X_2$ themselves.

$S$ should rather be called $S_n$ as it varies with $n$ and "approach" should be made precise with "converges in distribution to (-)".

$S$ is random because it depends on $\bar{X_n}$, which depends on the $X_1,\cdots,X_n$ which are random.

The relation to sample sets is that if you take quite a large set of independent samples from your population and computed its mean, you would heuristically expect the mean value to follow a certain distribution (regardless of the distribution of the samples).