Normal linear model for the response

The most commonly used regression model is a normal linear model. It involves:

Normality
At each value of X, Y has a normal distribution.
Constant variance
The standard deviation of Y is the same for all values of X.
Linearity
The mean of Y is linearly related to X.

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.