# Regression's population parameters

Suppose I've specified a linear regression model: $$Y = \beta_0 + \beta_1 X + \epsilon$$ where $$\beta_0$$, $$\beta_1$$ are the population parameters. My question is: why are these parameters populational? My intuition is that they are calculated based on all the possible realizations of the random variables $$X$$ and $$Y$$. The coefficient $$\beta_1$$ is $$E(XX')^1E(XY)$$ where the expected values ​​are calculated based on all the possible realizations of both variables and their probabilities.

The linear regression model that you specified can be view as a linear approximation of expected value of $$Y$$ on $$X$$. In many presentation about regression this fact is not clearly affirmed but it seem me very relevant. So, as usual in mathematical statistics, you have some parameters $$\theta$$s that characterize the joint distribution $$(Y,X)$$. These are population parameters or, equivalently, the exact/true parameters. You can see $$\beta$$s as transformations of $$\theta$$s, so you can see the $$\beta$$s as population parameters also.
Changing point of view we can think about statistical inference. So, very briefly, we can collect data from $$Y$$ and $$X$$ and estimate a regression in order to analyze some associations of interest (note: estimated approximation of expected value of $$Y$$ given $$X$$). Here you achieve the estimated parameters $$\beta_{est}$$ that seem me what you have in mind (est stand for estimator). Moreover you can interested in many features of $$\beta_{est}$$, for instance efficiency. In fact $$\beta_{est}$$ have its variance while $$\beta$$ is a constant. This happen precisely because $$\beta$$ collect population parameters and $$\beta_{est}$$ their estimators.