From the probability and statistics books, that i have studied, my understanding of central limit theorem is as follows:
CLT is applicable in case of repeated samples i.e. when we take a single sample of n observations and then we repeat the same experiment a number Of times to generate a group of samples and each such sample is of size $n$. So if the sample means of each such sample is plotted it will tend to follow a normal dist provided n is large say at least $>= 30$.
But using probability theory when we state CLT as:
Let $X_1, X_2,\ldots X_n$ is an iid random sample with mean mu and variance sigma-square. So the CLT is stated as $$ \frac{(\sum_iX_i)-n\mu}{\sigma \sqrt n}$$ tends to $N(0,1)$. So it is the cdf that tends to normal cdf and not pdf even if the original distribution is Non-normal
So, my question is how the concept of repeated samples, as defined for CLT, is implemented uaing random variable definition of CLT?
My understanding is that $X_i$ represent the $i$-th observation among the n observations of a single random sample.
Or, is it that, Here, $n$ does not define number of observations in a single sample rather it denotes no. Of samples taken where each sample is of equal size and $X_i$ represent some measurable sample characteristic of each such sample, say sample mean?
Please clarify?
The $X_i$ are random variables and each one $X_1,X_2,$ etc. represents the value of a particular observation. $n$ is the number of observations taken, as in $30$ in your example.
The CLT as you stated says that (under suitable conditions) $$ \sqrt{n}\frac{\bar X-\mu}{\sigma} \to_D N(0,1)$$ so that the sample mean's distribution begins to resemble the distribution of a $N(\mu,\sigma^2/n)$ as $n\to \infty.$
This says the exact same thing as you said colloquially before. If we did the experiment where we took $100$ observations a bunch of times and plotted a distribution of the resulting sample means then (provided $100$ is large enough for the particular application) it would look approximately normal with mean $\mu$ and variance $\sigma^2/n.$