Samples and random variables Suppose I picked a sample of $n$ 20-year-olds. I measured the height of each to obtain $n$ numbers: $h_1, h_2, \ldots, h_n$. According to theory of probability/statistics, there are $n$ random variables associated with the sample, say $X_1, X_2, \ldots, X_n$. However, I do not understand the relationship between the $X_i$ and the $h_i$, I have yet to see it explained clearly in any book and so I have a few questions.


*

*What is the probability space corresponding to the $X_i$? It seems to me that the way one samples should effect what the probability space will look like. In this case, I am sampling without replacement and the order in which I picked the individuals is irrelevant so I believe the sample space $\Omega$ should consist of all $n$-tuples of 20-year-olds such that no two tuples contain the same individuals. In this way, $X_i(\omega)$ is the height of the $i$th individual in the $n$-tuple $\omega \in \Omega$. The sample I picked would therefore correspond to one particlar point in $\Omega$, call it $\omega_0$, such that $X_i(\omega_0) = h_i$. I surmise that the $\sigma$-algebra will be just the power set $2^\Omega$ of $\Omega$ but I haven't a clue as to what the probability measure would be.

*Let $(\Gamma, 2^\Gamma, P)$ be a probability space where $\Gamma$ is the set of all 20-year-olds and let $X$ be a random variable on $\Gamma$ such that $X(\gamma)$ is the height of the individual $\gamma\in\Gamma$. What is the connection between the $X_i$ and $X$ besides that afforded to us by the law of large numbers? In particular, what is the exact relationship between the probability space of $X$ and that of the $X_i$?

Your experiment consists of choosing $n$ people from a certain population of 20-year-olds and measuring their heights.  $X_i$ is the height of the $i$'th person chosen.  In the particular outcome you obtained when you did this experiment, the value of $X_i$ was $h_i$.
The sample space $\Omega$ is all ordered $n$-tuples of distinct individuals from the population.
Since the sample space is finite (there being only a finite number of 20-year-olds in the world), the $\sigma$-algebra is indeed $2^\Omega$.  If you choose your sample "at random", the probability measure assigns equal probabilities to all $\omega \in \Omega$.
