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From measure theoretic foundations, it is clear that a random variable is neither random nor a variable. It is a deterministic function developed as follows:

Construct probability space: $(\Omega, \Sigma, \mathbb{P})$. Construct a measurable space: $(E, \mathcal{E})$. Then define random variable $X: \Omega \rightarrow E$, so that $X$ is a deterministic function parameterized as $X(\omega\in\Omega)$.

It is thus clearly a rather deterministic (non-random) function.

I'm thus curious as to its first official labelling as being a random variable. Is there a source or person who coined the term, and who also clearly justified as to why it should be thought as such?

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    $\begingroup$ I wonder why math/stats/ML folks have a tendency to use the word “clearly” before a controversial or insufficiently defended opinion, as if it gives more weight. $\endgroup$
    – Arya McCarthy
    Commented Aug 10, 2021 at 15:34
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    $\begingroup$ Omega is the sample space. It is the act of sampling from Omega that provides the randomness. X maps Omega to the real line. Correct me if I am wrong. If I sample an omega in Omega, this maps to X=x. $\endgroup$
    – Geoffrey Johnson
    Commented Aug 10, 2021 at 15:36
  • $\begingroup$ It is our secret lingo ;) I guess it's similar to how we use "trivial" as a meaning for the "base case" but we don't mean the the problem is necessarily "trivial" in the everyday (somewhat derisive?) sense of the term :) $\endgroup$
    – tisPrimeTime
    Commented Aug 10, 2021 at 15:37
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    $\begingroup$ @GeoffreyJohnson Seeing as how the word "random" doesn't really appear in the original probability space (and most of probability can be done without access to the random variable -- really it's more of a modelling convenience factor), why are we inspired to reserve the word "random" for the "random variable", and not reference in any of the prior measure-theoretic constructions? $\endgroup$
    – tisPrimeTime
    Commented Aug 10, 2021 at 15:39
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    $\begingroup$ I think $\Omega$ has nothing to do with randomness it just defines all the possible outcomes of the experiment. The fact that $X$ can take a variety of different outcomes with a certain probability I think is what it makes it random. stats.stackexchange.com/questions/139989/… $\endgroup$
    – Fiodor1234
    Commented Aug 10, 2021 at 16:02

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As to who coined the term, I suggest checking out the two below posts:

https://hsm.stackexchange.com/questions/9716/who-coined-the-term-random-variable

https://hsm.stackexchange.com/questions/2223/who-introduced-random-variables-into-probability

And a helpful page on getting around the precise meaning of "Random":

https://towardsdatascience.com/but-what-is-a-random-variable-4265d84cb7e5

You see, the events corresponding to your experiment have inherent uncertainty (randomness) associated with it i.e. your two coin toss in above experiment could be HH or HT or TT or TH. You then use probability theory to quantify the uncertainty corresponding to these events. I appreciate that at the end of the day it is simply semantics but I really liked the word uncertainty as it helps me not bring in my understanding of randomness from other disciplines. This also means that Random Variables in statistics could have been called Uncertain Variables. But they are not called so :( ….. the literature consistently calls them Random Variable so if it helps, you could (as I often) do the translation in your mind to Uncertain Variables.

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  • $\begingroup$ Thanks for the links on the origins of its first usage! They seem invaluable. As to the main question -- I think those origin papers don't really go into detail why they coined as "random variable" only that "at this point in time it began to be coined as such". I'm curious on this point because as mentioned the construction of this term seems to be motivated as a non-deterministic function (not as anything "random" nor a "variable"). --- However I'm not sure if it is possible to find such an answer, and we may need to "make an educated guess or two" as to why this is so? $\endgroup$
    – tisPrimeTime
    Commented Aug 10, 2021 at 21:13
  • $\begingroup$ @tisPrimeTime I definitely don't know the definitive reasoning behind the use of "random" in the random variable title, but in consideration of one of the definitions of random as: "made, done, happening, or chosen without method or conscious decision", my slightly-educated guess would be to think of the lack of "method or conscious decision" as there is no process determining the sampled values. For a fair coin flip, there is that underlying distribution of 50% heads and 50% tails over the long run, but any one sample the proportion of heads or tails doesn't have to follow these proportions $\endgroup$
    – jros
    Commented Aug 12, 2021 at 11:47

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