I have a basic doubt in sample and random variables. I have read related posts on this site but still some doubt is still left.
Suppose we have a population and we are drawing some entries from it which is called as sample. Suppose we have n entries then we say that "we have drawn a sample of size n".
Now, the doubt begins when we try to associate the random variable with it.
Suppose in the first experiment we draw a sample of size "n". The corresponding random variable is $X_1$ where the range($X_1$) is the sample itself.
Suppose we have done the experiment $k$ times, so we have $X_1,\;X_2,\;...,\;X_k$ random variables each corresponds to a sample of size $n$.

This is the conclusion I have made so far after reading many related posts.

Now I am reading a book on statistics and there it is written that,

Let $X_1,\;X_2,\; . . . ,\;X_n$ be i.i.d. random variables. The “empirical distribution” based on these is the discrete distribution with probability mass function given by
$f(t) = \frac{1}{n} \#{\{X_i=t}\}$
The sample mean of these is $\bar X = \frac{X_1+X_2+...+X_n}{n}$
$\bar X$ is also a random variable.

So from this I can infer that each $X_i$ corresponds to a particular sample from the population not the $n$ values of a random variable $X$.
But in the definition of empirical distribution there is no emphasis on the size of the sample corresponding $X_i$. Has it been assumed each sample also has size $n$ because if each sample has size $k\neq n$ then there should be $k$ in the denominator of the empirical distribution?

Also here the sample mean is basically a random variable. But by terminology, one can thought of sample mean as mean of a sample that is expectation of a particular random variable in the above context that sample mean, i.e., $\frac{\sum_j X_{ij}}{k}$. But here sample mean is a random variable. I have seen the same terminology in other books also as used in this book. Isn't they make the concepts confusing. I think there should be better term for this instead of sample mean. Or may be I am misunderstanding something.

After reading the comments, I have found precisely what is my doubt.
In Sheldon Ross's Probability and Statistics, it is written that

If $X_1,\;X_2,\;...,\;X_n$ are independent random variables having a common distribution $F$, then we say that they constitute a sample (sometimes called a random sample) from the distribution $F$.

I am slightly confused with definition of sample. Suppose our population is the annual income of a person lived in a particular city. So we take say 10 people from the population record their income (so we can attribute a correspondence between people and income as a random variable) denote it by $X_1$.
Now we put those $10$ people back in the population and again take $10$ people from it randomly denote by $X_2$. We have repeated it say $20$ times.
So we have $X_1,X_2,...,X_{20}$. I have a doubt that would we call it the $20$ samples of $10$ people each or just one sample of $200$ people (or more precisely the $200$ values of incomes)?

  • 3
    $\begingroup$ You go astray at the outset by not distinguishing the values of random variables from outcomes from samples. See stats.stackexchange.com/a/54894/919 for one explanation of the difference. A similar explanation at stats.stackexchange.com/a/96000/919 covers the sum of random variables, which (upon division by a constant) gives the sample mean. $\endgroup$
    – whuber
    Commented Feb 4, 2023 at 19:12
  • 1
    $\begingroup$ A couple of points to remember: (a) sampling without replacement from a finite population is not the same thing as sampling from a distribution (in the former, you do not have independence between the individual observations); (b) the sample mean can used to describe the mean of the $n$ values you have observed (and so is what it is), or the mean you might observe taking a sample sized $n$ (and so would be a random variable with a distribution) $\endgroup$
    – Henry
    Commented Feb 4, 2023 at 19:45
  • $\begingroup$ @whuber, thanks for sharing the train-in-a-model, it simplifies a lot of concepts. I have one doubt. So basically in a box we have tickets, they are basically "population". Now suppose we are drawing "$k$" tickets with replacement as our first experiment denote that sample by random variable $X_1$. We do this process say $n$ times. So we have $k$ random variables, these random variables are actually same (as the assignment is same, it is just that the values we get in the $n$ experiments are different) $\endgroup$
    – Iti
    Commented Feb 5, 2023 at 4:12
  • $\begingroup$ As given in the question, the empirical distribution for any $X_i$, will it be $f(t)=\frac{1}{k}#{\{Xi=t}\}$. But in the definition, given in my question, the size of each sample is irrelevant? Or are they considering implicitly that each $X_i$ corresponds to a sample of size $n$? $\endgroup$
    – Iti
    Commented Feb 5, 2023 at 4:14
  • 1
    $\begingroup$ I will repeat: a sample is not a random variable! $\endgroup$
    – whuber
    Commented Feb 5, 2023 at 14:55


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