“Sample of n random variables” is a simplified way of saying that

we have a sample drawn from the population, that we assume to be n identically distributed random variables

. So such sample behaves like n random variables. It’s ambiguous because it mixes terminology used in probability and statistics.

Solution 1:

The measurable function $(X^1,ldots,X^N)$ is a sample from $Omega_1$ to $Omega_2^N$. At $omegainOmega_1$, the value taken by the function $(x^1,ldots,x^N)=(X^1(omega),ldots,X^N(omega))$ is considered as a realization of this sample.

When stating

Assuming that $X^n$ represents functions and $X^n$ is equal to $X$.

The distinct nature of the functions $X^n$ becomes evident as the images $X^1(omega),ldots,X^N(omega)$ may vary for a given $omega$. In the case of iid samples, the functions $X^n$ possess two additional characteristics that set them apart.

- The distribution of $X^1, X^2, dots, X^N$ is the same, indicating that the probability of any measurable set $A$ in $mathcal{F}_2$ containing $X^1$ is equal to the probability of $A$ containing $X^2$, and so on up to $X^N$.
- The concept of independence is expressed as $mathbb{P}(X^1in A^1,ldots,X^Nin A^N)=mathbb{P}(X^1in A^1)cdotsmathbb{P}(X^Nin A^N)$ for any measurable sets $A^1,ldots,A^N$ in $mathcal{F}_2$.

Your definition

The expected value of X can be calculated by integrating X(ω1) over the domain Ω1.

is incorrect: it should be

The expected value of X can be calculated by integrating X(ω1) over the sample space Ω1 with respect to the probability measure P(ω1).

Solution 2:

A sample is drawn from a population, not from a random variable. Referring to a “sample of $n$ random variables” is a simplified way of saying that we have a sample drawn from the population, which we assume to be $n$ identically distributed random variables. This terminology can be ambiguous, as it mixes probability and statistics terminology. Similarly, in simulations, samples are drawn from a common distribution, and the resulting sample is considered as data. Samples are treated as random variables because they are drawn from random processes and are identically distributed since they come from the same distribution. Statistics are used to analyze samples, which use abstract, mathematical descriptions of problems in terms of probability theory. Random variables are functions that assign probabilities to events that can occur in your samples.