What does "fixed but random" mean exactly? I am studying econometrics and bumped into such thing as "fixed but random" variables. As far as I understood, these are the observations that will become constant values in the future, when they are observed. But until they are not, so while we are "theorising", they can be seen as random variables, e.g., they can be iid from each other, so have their own distribution.
That's how I understood the concept, at least. So, we know that they aren't fully random variables: we know that each of them is a separated number, but we don't know which number exactly, so it is yet random. A random observation, so to say. And it has a distribution. Or is it the underlying random variable that has the distribution?
Sorry, I think I got confused. Especially because different textbooks and professors explain it differently and use different notations. I would be more than grateful if you provide any explanation or at least links to proper literature.
Sorry if it's not the proper place, but I don't know where else to ask. When I try to google "fixed but random", it results in distinguishing between fixed and random effects of panel data.
Thank you in advance!
EDIT -------------
First of all, thank you so much for taking time to comment and answer me!

*

*About the term itself:

I have heard the "fixed but random" term from only one of my professors, and have never seen it anywhere else. I just thought that it is how these things are called among those who are deeply into econometrics or something. It is definitely not from Bayes statistic, as I haven't yet got that far.


*About what I actually do not understand

I will try to describe a more precise situation for which I thought the term "fixed but random" is used (now, thanks to you, I understand that I was wrong).
I will now be "quoting" the lectures of one of my professors. They assume the case of linear regression model and write it like this:
$Y = b_1X_1 + ... + b_nX_n + e$
Then they say that in terms of the sample this model corresponds to:
$y_i = x_i^T b +e_i$
i = 1, ..., n,
where x_i and b are vectors.
And then they state the assumptions for the model like this:
the process/sequence
$(y_i , x_i^T)$
i ∊ ℕ
is iid.
And this is where I get confused. As far as I understand, the random variables are those Y and X_1, ..., X_n of the initial model. And y_i and x_i are a single (i-th) observation of Y and a vector of (i-th) observations of X respectively. So, they are fixed. But here comes a question: how can observations be iid from each other if they are fixed, so they don't have their own distribution, they are just constants.
That's why I thought that such observations are called "fixed but random", because we consider them to be not yet observations (we look at the distribution from which they occur), but not random variables, too, since we know that they are supposed to be fixed numbers. Probably I am somewhere wrong here. I would be very grateful if you could provide the explanation. I hope I described the situation more clearly now.
Thank you!
EDIT 2: I tried to ask it as a separate question but it got closed with some links to questions like "what is a random variable?" I understand how rv's differ from non-random values. I wanted explanation for this specific setting and formulation. So I will leave the edition here. Maybe someone will answer.
 A: "fixed but random" is not a common term with a clear consensus about it's meaning.
The occurance of the term in the English literature is not common and half the time it is part of the phrase "not fixed but random"

Note that 'fixed but random' is not a contradiction when 'fixed' and 'random' relate to different things. But which relations, that depends on the context.

*

*Searching for the term on Google scholar I see several cases where 'fixed' refers to the situation where an object of study is fixed. And the 'random' part refers to the situation where this fixed state has been generated random.
For instance here

We are in a typical situation of the quenched disorder [3,4] when the motion of particles in a fixed but random environment is considered. That is why inevitably we are
faced with an averaging procedure, which helps us to pass to the macroscopic transport equation. $$$$ We have a fixed realization $b(x)=b(x_n)=b_n$ of a random
process (well depths)  with the probability distribution $P(b)$



*A situation of 'fixed but random' could also be the situation with Bayesian statistics where we use a prior distributions to express the probability distribution of some value which in reality may be a fixed value.


*Another situation of 'fixed but random' could be in description of random number generators. These can come up with a fixed sequence every time, but the numbers have a random character.
So this combination 'fixed but random' may occur and the meaning depends on the context. What it means for your econometrics case is not clear (since it is broad). It requires the specific context in order to say anything more about it.
A: This answer is very hand-wavy but hopefully captures the intuition of "fixed but random."
We sometimes assume that the observations on the explanatory variables can be considered fixed. For example, if Framer Ron is studying fertilizer's effect on turnip yield, he fixes the value of the manure dummy to one when he applies the fertilizer to a plot and zero when he does not. Then all the randomness in crop yield comes from unobserved characteristics of the plot, like how close the rabbit warren is (assuming rabbits steal turnips).
The next level is fixed in repeated samples. That is, it is possible to redraw the sample with the same explanatory variable values. This is often weak­ened to read that the explanatory variables are random but independent of the error term. For example, if I sample people from all over Canada, there are ten provinces and three territories. If you imagine there is some unknown province/territory-level effect, if you sample enough people from each, you can get a good handle on those 13 effects using dummy variables. In this case, it often makes sense to think of each unit-level effect $\alpha_i$ as an unknown parameter (so random to the analyst), but fixed once you look at enough data since there are only so many provinces. This is most likely what "fixed but random" means. An extreme example of this is a census, where the sample is the population.
This is in contrast to a panel data set where you sample units from some population. A consumer panel is one example, where you track a sample of households and their shopping over time. In that case, the household-level effect $u_i$ is not fixed since each time you sample a household you get a completely new $u_i$ draw. Here repeated sampling does not help you pin down the household effect, so it is in some sense more random and not fixed.
A: There is no such thing as something being simultaneously "fixed but random" --- those are contradictory concepts and this appelation is not used in statistics or econometrics so far as I am aware.  (If you can provide a source for the occurrence of this appelation then I'm happy to look at it.)  In classical statistics we sometimes refer to parameters as "unknown constants" (i.e., fixed but unknown) but we wouldn't refer to something as "fixed but random".  We also often take quantities that are fixed but unknown (fixed in an aleatory sense) and treat these as random variables in an epistemological sense because they are unknown.
While there is no such thing as something being fixed and random at the same time, in probability and statistics we often consider random variables that are considered to be random prior to occurrence or observation, but fixed after they become manifest or observed.  This means that problems frequently feature instances of a random variable (sometimes denoted in upper-case) and corresponding instances of its outcome (denoted in lower-case).  When you learn probability and statistics you frequently have to get used to seeing these upper and lower-case instances of the same variable, with one representing the pre-observational random variable and the other representing the post-observational fixed value.
Since you have not given a source or context for the notion of "fixed but random" my suspicion is that it is just an error, or a clumsy attempt to describe something else.  Without further context it is difficult to guess what this might be intended to describe.
