# Regression: What is the difference between assuming the covariates are random or not random? [duplicate]

I often see regression expressed in two ways.

1. The covariates are random: In this scenario, we have $$(x_i,y_i) \sim G$$ for some distribution $$G$$ and are i.i.d. for $$i = 1, \cdots, n$$. We then posit $$y_i \mid x_i \sim A$$ for some distribution $$A$$, and we want to successfully parameterize $$A$$ using maximum likelihood or some other method.
2. The covariates are fixed: In this scenario we have associated points $$(x_i, y_i)$$. We assume $$y_i \sim A$$ for some distribution $$A$$, and we assume $$x_i$$ parameterizes the distribution $$A$$ in some way. We then use maximum likelihood to find the best values of the other parameter(s) of $$A$$.

My question is, is this distinction between random covariates $$x_i$$ or non-random covariates $$x_i$$ important? At the end of the day it seems that $$x_i$$ figures in the pmf/pdf of our distribution $$A$$ anyhow.