I am learning about the generation of random numbers, however, many references instead talk about how to generate random variables.
Many references will write something like "Suppose we want to generate random variables $X_1, X_2, \ldots, X_n \in \mathcal{F}(S)$", where $\mathcal{F}(S)$ is the function space of random variables, $S$ is the underlying event space.
But from the context, what they actually mean is to generate random numbers $x_1, x_2, \ldots, x_n \in \mathbb{R}^n$.
Are these (hundreds if not thousands of references) confusing the meaning of the random variable with the realization of the random variable? Should I think of $X_1$ as a number or as a function?
Can someone please explain the reasoning behind using the phrase "to generate a random variable" instead just saying "to generate random numbers"?
Examples:
https://www.win.tue.nl/~marko/2WB05/lecture8.pdf http://opim.wharton.upenn.edu/~sok/papers/s/rv.pdf Generate random variable with given pdf https://www.encyclopediaofmath.org/index.php/Generating_random_variables http://math.ubbcluj.ro/~tradu/Randvargen.pdf
"Generating a random variable" is short for "implementing a random number generator that follows as specified distribution". Such a generator is usually built on top of an available generator of a uniformly distributed (pseudo-)random variable, via a transformation function or an algorithm.
So it is more a recipe to emulate a random variable than a set of drawings.