How does the distribution of the error term affect the distribution of the response?

So when I assume that the error terms are normally distributed in a linear regression, what does it mean for the response variable, $y$?

Answer

Maybe I’m off but I think we ought to be wondering about $f(y|\beta, X)$, which is how I read the OP. In the very simplest case of linear regression if your model is $y=X\beta + \epsilon$ then the only stochastic component in your model is the error term. As such it determines the sampling distribution of $y$. If $\epsilon\sim N(0, \sigma^2I)$ then $y|X, \beta\sim N(X\beta, \sigma^2I)$. What @Aniko says is certainly true of $f(y)$ (marginally over $X, \beta$), however. So as it stands the question is slightly vague.

Attribution
Source : Link , Question Author : MarkDollar , Answer Author : JMS

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