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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

Cettt
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Shamisen Expert
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    I think you are right in suspecting that this is an abuse of terminology. They really mean random numbers from a given random variable. – Fomalhaut Mar 13 '19 at 03:22
  • And then there is still the philosophical problem of what it means to generate a random number from a distribution. – Mark Mar 13 '19 at 03:24
  • @TomislavOstojich But there are literally thousands of references online that are saying the same thing, there must be logical reason behind it that I am not seeing? Or are these people all parroting off of one another – Shamisen Expert Mar 13 '19 at 03:25
  • @ShamisenExpert sometimes abuse of terminology becomes really popular. Like treating dy/dx as a fraction. Everyone does it and everyone knows it's wrong. – Fomalhaut Mar 13 '19 at 03:27
  • @TomislavOstojich Ok. I am haven't done statistics and only learned probability from an engineering context. I guess I haven't been indoctrinated yet – Shamisen Expert Mar 13 '19 at 03:27
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    @Mark Yes, what does it mean to generate a random number? That's another problem that I had. If I were to generate a single random number from a fixed distribution, you can almost never tell what that distribution is, but it is still generated from it. How many numbers are needed to get a sense of what this distribution looks like? – Shamisen Expert Mar 13 '19 at 03:37
  • @TomislavOstojich: using the modern definition of the differential, $dy/dx$ is indeed an ordinary fraction. –  Mar 13 '19 at 09:26
  • Sample size determination is important when you want to know probability. Your question may be better answered in a Statistics forum. This may be of help: https://stats.stackexchange.com/questions/72678/minimum-sample-size-required-to-estimate-the-probability-px-le-c-for-a-cons – NoChance Mar 13 '19 at 09:31

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"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.

  • This is what "generating a random variable" would mean in a programming context but probability classes at universities don't care about implementation. In the context of university class it probably means "a drawing" or "a set of drawings." – Fomalhaut Mar 14 '19 at 01:49