Statistical models and likelihood functions

[Cinitiator]

Homework Statement

I have a couple of notation interpretation questions:
1) What does f_X(x|θ) represent in this case? The realization function of of our random vector X for some value x and a parameter θ (so that if our random vector has n random variables, its realization vector will be a subset of R^n)? Or is something else represented here?

2) If our (non-parametrized) statistical model is based on some random vector X with n random variables, will it contain realization functions of the random vector, or rather the random variable functions which the said vector contains?

3) In case of parametrized models: Is the statistical model set (let's name it P) a set of functions under every parameter space possible? And what do these functions represent? Are they assumed to have a fixed input? Are they realizations of a random vector under every single parameter in the parameter space? Or are they random variable functions which belong to our random vector?

Homework Equations

The Attempt at a Solution

Trying to interpret it in different ways, but not knowing which interpretation is the correct one.

 

[haruspex]

1) What does f_X(x|θ) represent in this case?

It is the probability (density?) function for the r.v. X given the observations θ.
I don't know what you mean by a realization function.

 

[Cinitiator]

It is the probability (density?) function for the r.v. X given the observations θ.
I don't know what you mean by a realization function.

I know that. I don't know in what precise way this function is 'generated'. That is - do we take an entire random vector (say, an R^n vector with random variables) and input it, and then output a R^n vector in the measure space (probabilities for each R^...)? That's what I call a realization function, since it outputs the realization of this random vector.

Or is the said function generated in an entirely different way given our random variable?

 

[haruspex]

do we take an entire random vector (say, an R^n vector with random variables) and input it, and then output a R^n vector in the measure space (probabilities for each R^...)?

No, it can't be that. If you look at the definition of the likelihood function you can deduce that the range of f is ℝ, not ℝⁿ. So I would say it's just a joint distribution.