# Why use residual plots in linear regression for assessing normality?

Let's take the case of condition simple linear regression for example where we are assuming: $$Y|X=x = \beta_0 + \beta_1 x + \epsilon,$$ where $$\epsilon$$ represent the random noise.

In order to conduct inference my model assumptions are (1) the existence of a linear relationship between my predictor and my response, (2) that the random noise in my observations of my response has mean 0 and constant variance and that none of the noise is dependent, and (3) that the noise is Gaussian. Furthermore, we know that the residuals $$e_i$$ based on our model are not actually observations of the random noise.

My question is therefore, when checking for the third assumption, why do we only look at a Normal probability plot of the residuals? Why not instead use a Normal probability plot of the observed $$Y$$ values?

And as a followup question, suppose we assume the errors are distributed according to a non-Gaussian distribution. Are there any examples where if the random noise is non-Gaussian then the conditional response, $$Y|X=x$$, could follow a different distribution than the errors? (I.e. if $$\epsilon \sim \cal{P}$$ for some general probability distribution $$\cal{P}$$, are there any examples where the transformation $$Y = const. + \epsilon$$ is not also distributed as $$\cal{P}$$?

1. Usually you assume that the conditional distribution of $$Y$$ given $$X$$ is normal, hence checking the distribution of $$\{y_i\}_{i=1}^n$$ is checking the unconditional distribution of $$Y$$, and checking the distribution of $$\{e_i\}_{i=1}^n$$ is the same as checking the conditional distribution of $$Y$$.