Normal linear model for the response
The most commonly used regression model is a normal linear model. It involves:
The last two properties of the normal linear model can be expressed as
σy = σ
μy = β0 + β1x
The diagram below illustrates these three properties of the normal linear model: the distributions at different x-values have normal distributions with the same spread and the mean increases linearly with x.
Note: only the response is modelled
A normal linear model does not try to explain the distribution of x-values. In experimental data, they are fixed by the experimenter. In observational data, the x-values are usually random, but the regression model only explains how the y-values are related to them and treats them as constants.
The regression model only describes the conditional distribution of Y at each X.