I'm an applied researcher and had a conceptual question about the simple linear regression population.
From some books, I know in simple linear regression population, we assume to have a "subpopulation" for each value of the predictor (see picture below). Each of these subpopulations, is assumed to be a Normal Distribution. All of these subpopulations are assumed to have the same $\sigma^2$. But each of these subpopulations may have a different $\mu$ provided that these differing $\mu$s can be perfectly linearly connected to each other (see my plot below where I show 4 subpopulations, all of which have the same $\sigma^2$ but with perfectly linearly related $\mu$s).
My Conceptual Question
So, each subpopulation is a normal distribution. BUT does that mean that the ENTIRE POPULATION (i.e., POPULATION = SUM of ALL Subpopulations; in the picture below, I'm showing 4 subpopulations of size $1000$, so suppose these 4 subpopulations together are my POPULATION of size $4000$) is also a normal distribution?