# Are feature values random variable realizations, or random variables themselves?

I'm new to ML/stats so got confused with what I supposed was simple notation.

For simplicity, say I have a data set with just one column:

From probabilistic perspective, I had understood that X represents a random variable, and each row value is an observation (i.e. realization) of this random variable. X then has a distribution, which let's assume is Gaussian.

I got confused however when I studied the Maximum Likelihood Estimate (MLE) method. In MLE literature, it's assumed that the individual row values are iid, which then permits us to multiply their distributions:

Isn't iid however a property of random variables? In other words seems MLE is assuming that each row value by itself is a random variable with its own distribution. Doesn't this contradict the previous interpretation that X is a random variable and the row values are just specific realizations?

Yes, $$X$$ is a RV, and the listed values can be interpreted as its realizations. In MLE, we don't multiply their distributions. We again use $$X$$'s distribution but substitute our data points in it. $$p(x_i|\theta)$$ actually means $$p_X(x_i|\theta)$$, i.e. the value of PDF of $$X$$ at the specific point $$x_i$$ (something that represents some sort of probability measure). MLE just tries to maximize your probability of having those realizations.
And right, being iid is a property belonging to RVs although, not related to this discussion directly, technically constants are also RVs (with probability of equaling their value is $$1$$), and constants are also iid. But, here we're not talking about the iid-ness of constants/realizations. Each realization, $$x_i$$, is assumed to be drawn from the distribution of $$X_i$$ that has the same distribution with $$X$$. Consider the rows as grades of students $$i=1,2,3,4$$, i.e. $$X_i$$. Each student's grade can be thought of as a random variable. Having realizations of them doesn't change this.
This confusion arises in many other places, not just MLE; for instance, sample mean. Your sample mean is $$(100+200+300+400)/4=250$$. But, sample mean has also a distribution.