If the dependent variable is normally distributed for a fixed set of predictor values, then the residual values should be normally distributed with a mean of 0.
I have two questions based on the above paragraph:
First, I don't understand why the above paragraph must be true. Could someone please provide more details why the above must be true? Would be great if the answer clearly states what properties of random variables are being used to arrive at the above conclusion and any other critical assumptions not explicitly stated in the paragraph. For simplicity, please assume a simple linear regression.
The second question is I am not really sure what the phrase "...for a fixed set of predictor values.." mean. Does this mean that:
Suppose we have the following pairs of observations ($x$ is the predictor value, $y$ is the dependent value):
$$( x_1 , y_1) , (x_2,y_2), (x_3,y_3)$$
Does this mean that for each $x_i$, there will be corresponding predicted value $y_i$ and all of these $y_i$ will follow a normal distribution with possibly different means for $y_i$ but same variance, where $i \in [1,3]$?